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2 3linear equations ii

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math123a

The document discusses solving linear equations. It states that to solve linear equations, each side should be simplified first by separating the x-terms from the numbers. It then introduces the change-side change-sign rule - when moving terms to the other side of the equation, their signs change. An example problem is worked through step-by-step to demonstrate. It also discusses the opposite rule - that if -x = c, then x = -c. Finally, it notes that equations involving fractions can always be restated without fractions by finding the least common denominator.

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Linear Equations II
In this section we make some remarks about solving equations. Linear Equations II
In this section we make some remarks about solving equations. Recall that to solve linear equations we simplify each side of equation first. Then we separate the x's from the numbers. Linear Equations II
In this section we make some remarks about solving equations. Recall that to solve linear equations we simplify each side of equation first. Then we separate the x's from the numbers and extract the value of a single x. I. (Change-side-change-sign Rule) Linear Equations II
In this section we make some remarks about solving equations. Recall that to solve linear equations we simplify each side of equation first. Then we separate the x's from the numbers and extract the value of a single x. I. (Change-side-change-sign Rule) When gathering the x's to one side of the equation and the number terms to the other side, just move terms across to other side and switch their signs Linear Equations II
In this section we make some remarks about solving equations. Recall that to solve linear equations we simplify each side of equation first. Then we separate the x's from the numbers and extract the value of a single x. I. (Change-side-change-sign Rule) When gathering the x's to one side of the equation and the number terms to the other side, just move terms across to other side and switch their signs (instead of adding or subtracting to remove terms). Linear Equations II
In this section we make some remarks about solving equations. Recall that to solve linear equations we simplify each side of equation first. Then we separate the x's from the numbers and extract the value of a single x. I. (Change-side-change-sign Rule) When gathering the x's to one side of the equation and the number terms to the other side, just move terms across to other side and switch their signs (instead of adding or subtracting to remove terms). x + a = b x – a = b Linear Equations II
In this section we make some remarks about solving equations. Recall that to solve linear equations we simplify each side of equation first. Then we separate the x's from the numbers and extract the value of a single x. I. (Change-side-change-sign Rule) When gathering the x's to one side of the equation and the number terms to the other side, just move terms across to other side and switch their signs (instead of adding or subtracting to remove terms). x + a = b x – a = b x = b – a Linear Equations II change-side change-sign
In this section we make some remarks about solving equations. Recall that to solve linear equations we simplify each side of equation first. Then we separate the x's from the numbers and extract the value of a single x. I. (Change-side-change-sign Rule) When gathering the x's to one side of the equation and the number terms to the other side, just move terms across to other side and switch their signs (instead of adding or subtracting to remove terms). x + a = b x – a = b x = b – a x = b + a Linear Equations II change-side change-sign change-side change-signor
In this section we make some remarks about solving equations. Recall that to solve linear equations we simplify each side of equation first. Then we separate the x's from the numbers and extract the value of a single x. I. (Change-side-change-sign Rule) When gathering the x's to one side of the equation and the number terms to the other side, just move terms across to other side and switch their signs (instead of adding or subtracting to remove terms). x + a = b x – a = b x = b – a x = b + a Linear Equations II change-side change-sign change-side change-signor There are two sides to group the x's, the right hand side or the left hand side.
In this section we make some remarks about solving equations. Recall that to solve linear equations we simplify each side of equation first. Then we separate the x's from the numbers and extract the value of a single x. I. (Change-side-change-sign Rule) When gathering the x's to one side of the equation and the number terms to the other side, just move terms across to other side and switch their signs (instead of adding or subtracting to remove terms). x + a = b x – a = b x = b – a x = b + a Linear Equations II change-side change-sign change-side change-signor There are two sides to group the x's, the right hand side or the left hand side. It’s better to collect the x's to the side so that it ends up with positive x-term.
Linear Equations II Example A. Solve. a. –5 = x + 6
Linear Equations II Example A. Solve. a. –5 = x + 6 –6 –5 = x move +6 to the left, it turns into –6
Linear Equations II Example A. Solve. a. –5 = x + 6 –6 –5 = x –11 = x move +6 to the left, it turns into –6
b. 3(x + 2) – x = 2(–2 + x) + x Linear Equations II Example A. Solve. a. –5 = x + 6 –6 –5 = x –11 = x move +6 to the left, it turns into –6
b. 3(x + 2) – x = 2(–2 + x) + x simplify each side Linear Equations II Example A. Solve. a. –5 = x + 6 –6 –5 = x –11 = x move +6 to the left, it turns into –6
b. 3(x + 2) – x = 2(–2 + x) + x 3x + 6 – x = –4 + 2x + x simplify each side Linear Equations II Example A. Solve. a. –5 = x + 6 –6 –5 = x –11 = x move +6 to the left, it turns into –6
b. 3(x + 2) – x = 2(–2 + x) + x 3x + 6 – x = –4 + 2x + x 2x + 6 = –4 + 3x simplify each side Linear Equations II Example A. Solve. a. –5 = x + 6 –6 –5 = x –11 = x move +6 to the left, it turns into –6
b. 3(x + 2) – x = 2(–2 + x) + x 3x + 6 – x = –4 + 2x + x 2x + 6 = –4 + 3x 4 + 6 = 3x – 2x simplify each side move 2x and –4 to the other sides, change their signs Linear Equations II Example A. Solve. a. –5 = x + 6 –6 –5 = x –11 = x move +6 to the left, it turns into –6
b. 3(x + 2) – x = 2(–2 + x) + x 3x + 6 – x = –4 + 2x + x 2x + 6 = –4 + 3x 4 + 6 = 3x – 2x 10 = x simplify each side move 2x and –4 to the other sides, change their signs Linear Equations II Example A. Solve. a. –5 = x + 6 –6 –5 = x –11 = x move +6 to the left, it turns into –6
b. 3(x + 2) – x = 2(–2 + x) + x 3x + 6 – x = –4 + 2x + x 2x + 6 = –4 + 3x 4 + 6 = 3x – 2x 10 = x simplify each side move 2x and –4 to the other sides, change their signs Linear Equations II c. + x = 4 3 2 1 Example A. Solve. a. –5 = x + 6 –6 –5 = x –11 = x move +6 to the left, it turns into –6
b. 3(x + 2) – x = 2(–2 + x) + x 3x + 6 – x = –4 + 2x + x 2x + 6 = –4 + 3x 4 + 6 = 3x – 2x 10 = x simplify each side move 2x and –4 to the other sides, change their signs Linear Equations II c. + x = 4 3 move to the other side, change its sign 3 42 1 Example A. Solve. a. –5 = x + 6 –6 –5 = x –11 = x move +6 to the left, it turns into –6
b. 3(x + 2) – x = 2(–2 + x) + x 3x + 6 – x = –4 + 2x + x 2x + 6 = –4 + 3x 4 + 6 = 3x – 2x 10 = x simplify each side move 2x and –4 to the other sides, change their signs Linear Equations II c. + x = 4 3 move to the other side, change its sign 3 42 1 x = 2 1 4 3– Example A. Solve. a. –5 = x + 6 –6 –5 = x –11 = x move +6 to the left, it turns into –6
b. 3(x + 2) – x = 2(–2 + x) + x 3x + 6 – x = –4 + 2x + x 2x + 6 = –4 + 3x 4 + 6 = 3x – 2x 10 = x simplify each side move 2x and –4 to the other sides, change their signs Linear Equations II c. + x = 4 3 move to the other side, change its sign 3 42 1 x = 2 1 4 3 x = – 4 2 4 3– Example A. Solve. a. –5 = x + 6 –6 –5 = x –11 = x move +6 to the left, it turns into –6
b. 3(x + 2) – x = 2(–2 + x) + x 3x + 6 – x = –4 + 2x + x 2x + 6 = –4 + 3x 4 + 6 = 3x – 2x 10 = x simplify each side move 2x and –4 to the other sides, change their signs Linear Equations II c. + x = 4 3 move to the other side, change its sign 3 42 1 x = 2 1 4 3 x = x = 4 –1 – 4 2 4 3– Example A. Solve. a. –5 = x + 6 –6 –5 = x –11 = x move +6 to the left, it turns into –6
II. (The Opposite Rule) If –x = c, then x = –c Linear Equations II
II. (The Opposite Rule) If –x = c, then x = –c Example B. a. If –x = –15 Linear Equations II
Linear Equations II II. (The Opposite Rule) If –x = c, then x = –c Example B. a. If –x = –15 then x = 15
4 3 Linear Equations II II. (The Opposite Rule) If –x = c, then x = –c Example B. a. If –x = –15 then x = 15 b. If –x =
4 3 4 3 Linear Equations II II. (The Opposite Rule) If –x = c, then x = –c Example B. a. If –x = –15 then x = 15 b. If –x = then x = –
4 3 4 3 Linear Equations II Recall that an equation or a relation expressed using fractions may always be restated in a way without using fractions. II. (The Opposite Rule) If –x = c, then x = –c Example B. a. If –x = –15 then x = 15 b. If –x = then x = –
4 3 4 3 Linear Equations II Recall that an equation or a relation expressed using fractions may always be restated in a way without using fractions. Example C. The value of 2/3 of an apple is the same as the value of ¾ of an orange, restate this relation in whole numbers. II. (The Opposite Rule) If –x = c, then x = –c Example B. a. If –x = –15 then x = 15 b. If –x = then x = –
4 3 4 3 Linear Equations II Recall that an equation or a relation expressed using fractions may always be restated in a way without using fractions. Example C. The value of 2/3 of an apple is the same as the value of ¾ of an orange, restate this relation in whole numbers. 2 3 A = 3 4 G,We've II. (The Opposite Rule) If –x = c, then x = –c Example B. a. If –x = –15 then x = 15 b. If –x = then x = –
4 3 4 3 Linear Equations II Recall that an equation or a relation expressed using fractions may always be restated in a way without using fractions. Example C. The value of 2/3 of an apple is the same as the value of ¾ of an orange, restate this relation in whole numbers. 2 3 A = 3 4 G,We've LCD =12, multiply 12 to both sides, II. (The Opposite Rule) If –x = c, then x = –c Example B. a. If –x = –15 then x = 15 b. If –x = then x = –
4 3 4 3 Linear Equations II Recall that an equation or a relation expressed using fractions may always be restated in a way without using fractions. Example C. The value of 2/3 of an apple is the same as the value of ¾ of an orange, restate this relation in whole numbers. 2 3 A = 3 4 G, ( ) *12 We've LCD =12, multiply 12 to both sides, 2 3 A = 3 4 G II. (The Opposite Rule) If –x = c, then x = –c Example B. a. If –x = –15 then x = 15 b. If –x = then x = –
4 3 4 3 Linear Equations II Recall that an equation or a relation expressed using fractions may always be restated in a way without using fractions. Example C. The value of 2/3 of an apple is the same as the value of ¾ of an orange, restate this relation in whole numbers. 2 3 A = 3 4 G, 4 ( ) *12 We've 2 3 A = 3 4 G LCD =12, multiply 12 to both sides, II. (The Opposite Rule) If –x = c, then x = –c Example B. a. If –x = –15 then x = 15 b. If –x = then x = –
4 3 4 3 Linear Equations II Recall that an equation or a relation expressed using fractions may always be restated in a way without using fractions. Example C. The value of 2/3 of an apple is the same as the value of ¾ of an orange, restate this relation in whole numbers. 2 3 A = 3 4 G, 34 ( ) *12 We've 2 3 A = 3 4 G LCD =12, multiply 12 to both sides, II. (The Opposite Rule) If –x = c, then x = –c Example B. a. If –x = –15 then x = 15 b. If –x = then x = –
4 3 4 3 Linear Equations II Recall that an equation or a relation expressed using fractions may always be restated in a way without using fractions. Example C. The value of 2/3 of an apple is the same as the value of ¾ of an orange, restate this relation in whole numbers. 2 3 A = 3 4 G, 34 8A = 9G ( ) *12 We've 2 3 A = 3 4 G or the value of 8 apples equal to 9 oranges. LCD =12, multiply 12 to both sides, II. (The Opposite Rule) If –x = c, then x = –c Example B. a. If –x = –15 then x = 15 b. If –x = then x = –
Linear Equations II III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
Example D. Solve the equations Linear Equations II x = –6 4 3a. III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve. Example D. Solve the equations Linear Equations II x = –6 4 3 x = –6 4 3 )( 4 a. clear the denominator, multiply both sides by 4
Example D. Solve the equations Linear Equations II x = –6 4 3 x = –6 4 3 3x = –24 )( 4 a. clear the denominator, multiply both sides by 4 III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
Example D. Solve the equations Linear Equations II x = –6 4 3 x = –6 4 3 3x = –24 x = –8 )( 4 a. clear the denominator, multiply both sides by 4 div by 3 III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
Example D. Solve the equations x 4 3 1 2 6 5 4 3 Linear Equations II –+ = xb. x = –6 4 3 x = –6 4 3 3x = –24 x = –8 )( 4 a. clear the denominator, multiply both sides by 4 div by 3 III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
Example D. Solve the equations x 4 3 1 2 6 5 4 3 the LCD = 12, multiply it to both sides to clear the denominators ( ) *12 Linear Equations II –+ = x x 4 3 1 2 6 5 4 3–+ = x b. x = –6 4 3 x = –6 4 3 3x = –24 x = –8 )( 4 a. clear the denominator, multiply both sides by 4 div by 3 III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
Example D. Solve the equations x 4 3 1 2 6 5 4 3 the LCD = 12, multiply it to both sides to clear the denominators ( ) *12 3 Linear Equations II –+ = x x 4 3 1 2 6 5 4 3–+ = x b. x = –6 4 3 x = –6 4 3 3x = –24 x = –8 )( 4 a. clear the denominator, multiply both sides by 4 div by 3 III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
Example D. Solve the equations x 4 3 1 2 6 5 4 3 the LCD = 12, multiply it to both sides to clear the denominators ( ) *12 3 4 Linear Equations II –+ = x x 4 3 1 2 6 5 4 3–+ = x b. x = –6 4 3 x = –6 4 3 3x = –24 x = –8 )( 4 a. clear the denominator, multiply both sides by 4 div by 3 III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
Example D. Solve the equations x 4 3 1 2 6 5 4 3 the LCD = 12, multiply it to both sides to clear the denominators ( ) *12 3 4 2 3 Linear Equations II –+ = x x 4 3 1 2 6 5 4 3–+ = x b. x = –6 4 3 x = –6 4 3 3x = –24 x = –8 )( 4 a. clear the denominator, multiply both sides by 4 div by 3 III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
Example D. Solve the equations x 4 3 1 2 6 5 4 3 the LCD = 12, multiply it to both sides to clear the denominators ( ) *12 3 4 2 3 3x + 8 = 10x – 9 Linear Equations II –+ = x x 4 3 1 2 6 5 4 3–+ = x b. x = –6 4 3 x = –6 4 3 3x = –24 x = –8 )( 4 a. clear the denominator, multiply both sides by 4 div by 3 III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
Example D. Solve the equations x 4 3 1 2 6 5 4 3 the LCD = 12, multiply it to both sides to clear the denominators ( ) *12 3 4 2 3 3x + 8 = 10x – 9 Linear Equations II –+ = x x 4 3 1 2 6 5 4 3–+ = x b. x = –6 4 3 x = –6 4 3 3x = –24 x = –8 )( 4 a. clear the denominator, multiply both sides by 4 move 3x to the right and –9 to the left and switch signs. div by 3 III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
Example D. Solve the equations x 4 3 1 2 6 5 4 3 the LCD = 12, multiply it to both sides to clear the denominators ( ) *12 3 4 2 3 3x + 8 = 10x – 9 9 + 8 = 10x – 3x Linear Equations II –+ = x x 4 3 1 2 6 5 4 3–+ = x b. x = –6 4 3 x = –6 4 3 3x = –24 x = –8 )( 4 a. clear the denominator, multiply both sides by 4 move 3x to the right and –9 to the left and switch signs. div by 3 III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
Example D. Solve the equations x 4 3 1 2 6 5 4 3 the LCD = 12, multiply it to both sides to clear the denominators ( ) *12 3 4 2 3 3x + 8 = 10x – 9 9 + 8 = 10x – 3x 17 = 7x Linear Equations II –+ = x x 4 3 1 2 6 5 4 3–+ = x b. x = –6 4 3 x = –6 4 3 3x = –24 x = –8 )( 4 a. clear the denominator, multiply both sides by 4 move 3x to the right and –9 to the left and switch signs. div by 3 III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
Example D. Solve the equations x 4 3 1 2 6 5 4 3 the LCD = 12, multiply it to both sides to clear the denominators ( ) *12 3 4 2 3 3x + 8 = 10x – 9 9 + 8 = 10x – 3x 17 = 7x = x 7 17 Linear Equations II –+ = x x 4 3 1 2 6 5 4 3–+ = x b. x = –6 4 3 x = –6 4 3 3x = –24 x = –8 )( 4 a. clear the denominator, multiply both sides by 4 move 3x to the right and –9 to the left and switch signs. div by 3 div by 7 III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
(x – 20) = x – 27 100 15 100 45 Linear Equations II c.
(x – 20) = x – 27 100 15 100 45 multiply 100 to both sides to remove denominators [ (x – 20) = x – 27 100 15 100 45 ] * 100 Linear Equations II c.
(x – 20) = x – 27 100 15 100 45 multiply 100 to both sides to remove denominators [ (x – 20) = x – 27 100 15 100 45 ] * 100 1 1 100 Linear Equations II c.
(x – 20) = x – 27 100 15 100 45 multiply 100 to both sides to remove denominators [ (x – 20) = x – 27 100 15 100 45 ] * 100 1 1 100 15 (x – 20) = 45x – 2700 Linear Equations II c.
(x – 20) = x – 27 100 15 100 45 multiply 100 to both sides to remove denominators [ (x – 20) = x – 27 100 15 100 45 ] * 100 1 1 100 15 (x – 20) = 45x – 2700 15x – 300 = 45x – 2700 Linear Equations II c.
(x – 20) = x – 27 100 15 100 45 multiply 100 to both sides to remove denominators [ (x – 20) = x – 27 100 15 100 45 ] * 100 1 1 100 15 (x – 20) = 45x – 2700 15x – 300 = 45x – 2700 2700 – 300 = 45x –15x Linear Equations II c.
(x – 20) = x – 27 100 15 100 45 multiply 100 to both sides to remove denominators [ (x – 20) = x – 27 100 15 100 45 ] * 100 1 1 100 15 (x – 20) = 45x – 2700 15x – 300 = 45x – 2700 2700 – 300 = 45x –15x 2400 = 30x Linear Equations II c.
(x – 20) = x – 27 100 15 100 45 multiply 100 to both sides to remove denominators [ (x – 20) = x – 27 100 15 100 45 ] * 100 1 1 100 15 (x – 20) = 45x – 2700 15x – 300 = 45x – 2700 2700 – 300 = 45x –15x 2400 = 30x 2400 / 30 = x 80 = x Linear Equations II c.
(x – 20) = x – 27 100 15 100 45 multiply 100 to both sides to remove denominators [ (x – 20) = x – 27 100 15 100 45 ] * 100 1 1 100 15 (x – 20) = 45x – 2700 15x – 300 = 45x – 2700 2700 – 300 = 45x –15x 2400 = 30x 2400 / 30 = x 80 = x Linear Equations II c. d. 0.25(x – 100) = 0.10x – 1
(x – 20) = x – 27 100 15 100 45 multiply 100 to both sides to remove denominators [ (x – 20) = x – 27 100 15 100 45 ] * 100 1 1 100 15 (x – 20) = 45x – 2700 15x – 300 = 45x – 2700 2700 – 300 = 45x –15x 2400 = 30x 2400 / 30 = x 80 = x Linear Equations II c. (x – 100) = x – 1100 25 100 10 d. 0.25(x – 100) = 0.10x – 1 Change the decimal into fractions, we get
(x – 20) = x – 27 100 15 100 45 multiply 100 to both sides to remove denominators [ (x – 20) = x – 27 100 15 100 45 ] * 100 1 1 100 15 (x – 20) = 45x – 2700 15x – 300 = 45x – 2700 2700 – 300 = 45x –15x 2400 = 30x 2400 / 30 = x 80 = x Linear Equations II c. (x – 100) = x – 1100 25 100 10 multiply 100 to both sides to remove denominators d. 0.25(x – 100) = 0.10x – 1 Change the decimal into fractions, we get You finish it…
(x – 20) = x – 27 100 15 100 45 multiply 100 to both sides to remove denominators [ (x – 20) = x – 27 100 15 100 45 ] * 100 1 1 100 15 (x – 20) = 45x – 2700 15x – 300 = 45x – 2700 2700 – 300 = 45x –15x 2400 = 30x 2400 / 30 = x 80 = x Linear Equations II c. (x – 100) = x – 1100 25 100 10 multiply 100 to both sides to remove denominators d. 0.25(x – 100) = 0.10x – 1 Change the decimal into fractions, we get You finish it… Ans: x =160
Linear Equations II IV. (Reduction Rule) Use division to reduce equations to simpler ones.
Linear Equations II IV. (Reduction Rule) Use division to reduce equations to simpler ones. Specifically divide the common factor of the coefficients of each term to make them smaller and easier to work with.
14x – 49 = 70x – 98 Linear Equations II Example. E. Simplify the equation first then solve. IV. (Reduction Rule) Use division to reduce equations to simpler ones. Specifically divide the common factor of the coefficients of each term to make them smaller and easier to work with.
14x – 49 = 70x – 98 divide each term by 7 Linear Equations II Example. E. Simplify the equation first then solve. 14x – 49 70x – 98= 7 7 7 7 IV. (Reduction Rule) Use division to reduce equations to simpler ones. Specifically divide the common factor of the coefficients of each term to make them smaller and easier to work with.
14x – 49 = 70x – 98 divide each term by 7 Linear Equations II Example. E. Simplify the equation first then solve. 14x – 49 70x – 98= 7 7 7 7 2x – 7 = 10x – 14 IV. (Reduction Rule) Use division to reduce equations to simpler ones. Specifically divide the common factor of the coefficients of each term to make them smaller and easier to work with.
14x – 49 = 70x – 98 divide each term by 7 Linear Equations II Example. E. Simplify the equation first then solve. 14x – 49 70x – 98= 7 7 7 7 2x – 7 = 10x – 14 14 – 7 = 10x – 2x IV. (Reduction Rule) Use division to reduce equations to simpler ones. Specifically divide the common factor of the coefficients of each term to make them smaller and easier to work with.
14x – 49 = 70x – 98 divide each term by 7 Linear Equations II Example. E. Simplify the equation first then solve. 14x – 49 70x – 98= 7 7 7 7 2x – 7 = 10x – 14 14 – 7 = 10x – 2x 7 = 8x = x 8 7 IV. (Reduction Rule) Use division to reduce equations to simpler ones. Specifically divide the common factor of the coefficients of each term to make them smaller and easier to work with.
14x – 49 = 70x – 98 divide each term by 7 Linear Equations II Example. E. Simplify the equation first then solve. 14x – 49 70x – 98= 7 7 7 7 2x – 7 = 10x – 14 14 – 7 = 10x – 2x 7 = 8x = x 8 7 We should always reduce the equation first if it is possible. IV. (Reduction Rule) Use division to reduce equations to simpler ones. Specifically divide the common factor of the coefficients of each term to make them smaller and easier to work with.
There’re two type of equations that give unusual results. The first type is referred to as identities. Linear Equations II
There’re two type of equations that give unusual results. The first type is referred to as identities. A simple identity is the equation “x = x”. Linear Equations II
There’re two type of equations that give unusual results. The first type is referred to as identities. A simple identity is the equation “x = x”. This corresponds to the trick question “What number x is equal to itself?” Linear Equations II
There’re two type of equations that give unusual results. The first type is referred to as identities. A simple identity is the equation “x = x”. This corresponds to the trick question “What number x is equal to itself?” The answer of course is that x can be any number or that the solutions of equation “x = x” are all numbers. Linear Equations II
There’re two type of equations that give unusual results. The first type is referred to as identities. A simple identity is the equation “x = x”. This corresponds to the trick question “What number x is equal to itself?” The answer of course is that x can be any number or that the solutions of equation “x = x” are all numbers. Linear Equations II This is also the case for the any equation where both sides are identical such as 2x + 1 = 2x + 1,1 – 4x = 1 – 4x etc…
There’re two type of equations that give unusual results. The first type is referred to as identities. A simple identity is the equation “x = x”. This corresponds to the trick question “What number x is equal to itself?” The answer of course is that x can be any number or that the solutions of equation “x = x” are all numbers. Linear Equations II This is also the case for the any equation where both sides are identical such as 2x + 1 = 2x + 1,1 – 4x = 1 – 4x etc… An equation with identical expressions on both sides or can be rearranged into identical sides has all numbers as its solutions. Such an equation is called an identity.
There’re two type of equations that give unusual results. The first type is referred to as identities. A simple identity is the equation “x = x”. This corresponds to the trick question “What number x is equal to itself?” The answer of course is that x can be any number or that the solutions of equation “x = x” are all numbers. Example F. Solve. Linear Equations II 2(x – 1) + 3 = x – (– x –1) This is also the case for the any equation where both sides are identical such as 2x + 1 = 2x + 1,1 – 4x = 1 – 4x etc… An equation with identical expressions on both sides or can be rearranged into identical sides has all numbers as its solutions. Such an equation is called an identity.
There’re two type of equations that give unusual results. The first type is referred to as identities. A simple identity is the equation “x = x”. This corresponds to the trick question “What number x is equal to itself?” The answer of course is that x can be any number or that the solutions of equation “x = x” are all numbers. Example F. Solve. Linear Equations II 2(x – 1) + 3 = x – (– x –1) This is also the case for the any equation where both sides are identical such as 2x + 1 = 2x + 1,1 – 4x = 1 – 4x etc… expand 2x – 2 + 3 = x + x + 1 An equation with identical expressions on both sides or can be rearranged into identical sides has all numbers as its solutions. Such an equation is called an identity.
There’re two type of equations that give unusual results. The first type is referred to as identities. A simple identity is the equation “x = x”. This corresponds to the trick question “What number x is equal to itself?” The answer of course is that x can be any number or that the solutions of equation “x = x” are all numbers. Example F. Solve. Linear Equations II 2(x – 1) + 3 = x – (– x –1) This is also the case for the any equation where both sides are identical such as 2x + 1 = 2x + 1,1 – 4x = 1 – 4x etc… expand 2x – 2 + 3 = x + x + 1 simplify 2x + 1 = 2x + 1 An equation with identical expressions on both sides or can be rearranged into identical sides has all numbers as its solutions. Such an equation is called an identity.
There’re two type of equations that give unusual results. The first type is referred to as identities. A simple identity is the equation “x = x”. This corresponds to the trick question “What number x is equal to itself?” The answer of course is that x can be any number or that the solutions of equation “x = x” are all numbers. Example F. Solve. Linear Equations II 2(x – 1) + 3 = x – (– x –1) This is also the case for the any equation where both sides are identical such as 2x + 1 = 2x + 1,1 – 4x = 1 – 4x etc… expand 2x – 2 + 3 = x + x + 1 simplify 2x + 1 = 2x + 1 An equation with identical expressions on both sides or can be rearranged into identical sides has all numbers as its solutions. Such an equation is called an identity. two sides are identical
There’re two type of equations that give unusual results. The first type is referred to as identities. A simple identity is the equation “x = x”. This corresponds to the trick question “What number x is equal to itself?” The answer of course is that x can be any number or that the solutions of equation “x = x” are all numbers. Example F. Solve. Linear Equations II 2(x – 1) + 3 = x – (– x –1) This is also the case for the any equation where both sides are identical such as 2x + 1 = 2x + 1,1 – 4x = 1 – 4x etc… expand 2x – 2 + 3 = x + x + 1 simplify 2x + 1 = 2x + 1 So this equation is an identity and every number is a solution. An equation with identical expressions on both sides or can be rearranged into identical sides has all numbers as its solutions. Such an equation is called an identity. two sides are identical
At the opposite end of the identities are the “impossible” equations where there is no solution at all. Linear Equations II
At the opposite end of the identities are the “impossible” equations where there is no solution at all. An example is the equation x = x + 1. Linear Equations II
At the opposite end of the identities are the “impossible” equations where there is no solution at all. An example is the equation x = x + 1. This corresponds to the trick question “What number is still the same after we add 1 to it?” Linear Equations II
At the opposite end of the identities are the “impossible” equations where there is no solution at all. An example is the equation x = x + 1. This corresponds to the trick question “What number is still the same after we add 1 to it?” Of course there no such number. Linear Equations II
At the opposite end of the identities are the “impossible” equations where there is no solution at all. An example is the equation x = x + 1. This corresponds to the trick question “What number is still the same after we add 1 to it?” Of course there no such number. Linear Equations II If we attempt to solve x = x + 1 x – x = 1
At the opposite end of the identities are the “impossible” equations where there is no solution at all. An example is the equation x = x + 1. This corresponds to the trick question “What number is still the same after we add 1 to it?” Of course there no such number. Linear Equations II If we attempt to solve x = x + 1 x – x = 1 0 = 1we get which is an impossibility.
At the opposite end of the identities are the “impossible” equations where there is no solution at all. An example is the equation x = x + 1. This corresponds to the trick question “What number is still the same after we add 1 to it?” Of course there no such number. Linear Equations II If we attempt to solve x = x + 1 x – x = 1 0 = 1we get which is an impossibility. These equations are called inconsistent equations.
At the opposite end of the identities are the “impossible” equations where there is no solution at all. An example is the equation x = x + 1. This corresponds to the trick question “What number is still the same after we add 1 to it?” Of course there no such number. Example F. Solve the equation Linear Equations II 2(x – 1) + 4 = x – (– x –1) If we attempt to solve x = x + 1 x – x = 1 0 = 1we get which is an impossibility. These equations are called inconsistent equations.
At the opposite end of the identities are the “impossible” equations where there is no solution at all. An example is the equation x = x + 1. This corresponds to the trick question “What number is still the same after we add 1 to it?” Of course there no such number. Example F. Solve the equation Linear Equations II 2(x – 1) + 4 = x – (– x –1) If we attempt to solve x = x + 1 x – x = 1 0 = 1we get expand 2x – 2 + 4 = x + x + 1 which is an impossibility. These equations are called inconsistent equations.
At the opposite end of the identities are the “impossible” equations where there is no solution at all. An example is the equation x = x + 1. This corresponds to the trick question “What number is still the same after we add 1 to it?” Of course there no such number. Example F. Solve the equation Linear Equations II 2(x – 1) + 4 = x – (– x –1) If we attempt to solve x = x + 1 x – x = 1 0 = 1we get expand 2x – 2 + 4 = x + x + 1 simplify 2x + 2 = 2x + 1 which is an impossibility. These equations are called inconsistent equations.
At the opposite end of the identities are the “impossible” equations where there is no solution at all. An example is the equation x = x + 1. This corresponds to the trick question “What number is still the same after we add 1 to it?” Of course there no such number. Example F. Solve the equation Linear Equations II 2(x – 1) + 4 = x – (– x –1) If we attempt to solve x = x + 1 x – x = 1 0 = 1we get expand 2x – 2 + 4 = x + x + 1 simplify 2x + 2 = 2x + 1 2 = 1 So this is an inconsistent equation and there is no solution. which is an impossibility. These equations are called inconsistent equations.
Exercise. A. Solve for x using the switch-side-switch-sign rule. Remember to move the x’s first and get positive x’s. 1. x + 2 = 5 – 2x 2. 2x – 1 = – x – 7 3. –x = x – 8 4. –x = 3 – 2x 5. –5x = 6 – 3x 6. –x + 2 = 3 + 2x 7. –3x – 1= 3 – 6x 8. –x + 7 = 3 – 3x 9. –2x + 2 = 9 + x Linear Equations II x = 5 2x = – 7 = x B. Solve the following fractional equations by using the LCD to remove the denominators first. x = –2 5 3 10. x = –5 3 –4 11. = 4 3x 12. 2 –1 = 3 9x 13. 2 3 x = 3 –2 14. 2 –1 = 6 7x 15. 4 –3
Linear Equations II x 6 3 1 2 3 5 2 3–+ = x16. x 4 6 –3 1 8 –5 – 1– = x17. x 4 5 3 2 10 7 4 3+– = x18. x 8 12 –5 7 16 –5 + 1+ = x19. (x – 20) = x – 3 100 30 100 2020. (x + 5) – 3 = (x – 5) 100 25 100 2021. (x +15) = x + 1 100 15 100 3522. (50 – x) + 2 = (x – 50) 100 25 100 2023. C. Reduce the equations then solve. 28. –3x – 12 = 30 – 6x 29. 15x – 10x = 25x – 20 30. –4(x – 3) = 12(x + 2) – 8x 31. 15x – 10(x + 2) = 25x – 20 24. –0.3x – 0.25 = 1 – 0.6x 25. 0.15x – 0.1x = 0.25x – 2.4 26. 0.37 – 0.17x = 0.19x – 0.1 27. 1.7x – 0.11 = 0.22 – 0.4x
Linear Equations II D. Identify which equations are identities and which are inconsistent. 34. –x + 1 = 5x – 2(3x + 1) 35. –2(x + 2) = 5x – (7x – 4) 36. 4(x – 3) – 2 = 1 – (14 – 4x) 37. 4(x – 3) – 2 = 1 – (15 – 4x) 32. –x + 1 = –x + 2 33. x + 2 = 5 – (3 – x)

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2 3linear equations ii

  • 2. In this section we make some remarks about solving equations. Linear Equations II
  • 3. In this section we make some remarks about solving equations. Recall that to solve linear equations we simplify each side of equation first. Then we separate the x's from the numbers. Linear Equations II
  • 4. In this section we make some remarks about solving equations. Recall that to solve linear equations we simplify each side of equation first. Then we separate the x's from the numbers and extract the value of a single x. I. (Change-side-change-sign Rule) Linear Equations II
  • 5. In this section we make some remarks about solving equations. Recall that to solve linear equations we simplify each side of equation first. Then we separate the x's from the numbers and extract the value of a single x. I. (Change-side-change-sign Rule) When gathering the x's to one side of the equation and the number terms to the other side, just move terms across to other side and switch their signs Linear Equations II
  • 6. In this section we make some remarks about solving equations. Recall that to solve linear equations we simplify each side of equation first. Then we separate the x's from the numbers and extract the value of a single x. I. (Change-side-change-sign Rule) When gathering the x's to one side of the equation and the number terms to the other side, just move terms across to other side and switch their signs (instead of adding or subtracting to remove terms). Linear Equations II
  • 7. In this section we make some remarks about solving equations. Recall that to solve linear equations we simplify each side of equation first. Then we separate the x's from the numbers and extract the value of a single x. I. (Change-side-change-sign Rule) When gathering the x's to one side of the equation and the number terms to the other side, just move terms across to other side and switch their signs (instead of adding or subtracting to remove terms). x + a = b x – a = b Linear Equations II
  • 8. In this section we make some remarks about solving equations. Recall that to solve linear equations we simplify each side of equation first. Then we separate the x's from the numbers and extract the value of a single x. I. (Change-side-change-sign Rule) When gathering the x's to one side of the equation and the number terms to the other side, just move terms across to other side and switch their signs (instead of adding or subtracting to remove terms). x + a = b x – a = b x = b – a Linear Equations II change-side change-sign
  • 9. In this section we make some remarks about solving equations. Recall that to solve linear equations we simplify each side of equation first. Then we separate the x's from the numbers and extract the value of a single x. I. (Change-side-change-sign Rule) When gathering the x's to one side of the equation and the number terms to the other side, just move terms across to other side and switch their signs (instead of adding or subtracting to remove terms). x + a = b x – a = b x = b – a x = b + a Linear Equations II change-side change-sign change-side change-signor
  • 10. In this section we make some remarks about solving equations. Recall that to solve linear equations we simplify each side of equation first. Then we separate the x's from the numbers and extract the value of a single x. I. (Change-side-change-sign Rule) When gathering the x's to one side of the equation and the number terms to the other side, just move terms across to other side and switch their signs (instead of adding or subtracting to remove terms). x + a = b x – a = b x = b – a x = b + a Linear Equations II change-side change-sign change-side change-signor There are two sides to group the x's, the right hand side or the left hand side.
  • 11. In this section we make some remarks about solving equations. Recall that to solve linear equations we simplify each side of equation first. Then we separate the x's from the numbers and extract the value of a single x. I. (Change-side-change-sign Rule) When gathering the x's to one side of the equation and the number terms to the other side, just move terms across to other side and switch their signs (instead of adding or subtracting to remove terms). x + a = b x – a = b x = b – a x = b + a Linear Equations II change-side change-sign change-side change-signor There are two sides to group the x's, the right hand side or the left hand side. It’s better to collect the x's to the side so that it ends up with positive x-term.
  • 12. Linear Equations II Example A. Solve. a. –5 = x + 6
  • 13. Linear Equations II Example A. Solve. a. –5 = x + 6 –6 –5 = x move +6 to the left, it turns into –6
  • 14. Linear Equations II Example A. Solve. a. –5 = x + 6 –6 –5 = x –11 = x move +6 to the left, it turns into –6
  • 15. b. 3(x + 2) – x = 2(–2 + x) + x Linear Equations II Example A. Solve. a. –5 = x + 6 –6 –5 = x –11 = x move +6 to the left, it turns into –6
  • 16. b. 3(x + 2) – x = 2(–2 + x) + x simplify each side Linear Equations II Example A. Solve. a. –5 = x + 6 –6 –5 = x –11 = x move +6 to the left, it turns into –6
  • 17. b. 3(x + 2) – x = 2(–2 + x) + x 3x + 6 – x = –4 + 2x + x simplify each side Linear Equations II Example A. Solve. a. –5 = x + 6 –6 –5 = x –11 = x move +6 to the left, it turns into –6
  • 18. b. 3(x + 2) – x = 2(–2 + x) + x 3x + 6 – x = –4 + 2x + x 2x + 6 = –4 + 3x simplify each side Linear Equations II Example A. Solve. a. –5 = x + 6 –6 –5 = x –11 = x move +6 to the left, it turns into –6
  • 19. b. 3(x + 2) – x = 2(–2 + x) + x 3x + 6 – x = –4 + 2x + x 2x + 6 = –4 + 3x 4 + 6 = 3x – 2x simplify each side move 2x and –4 to the other sides, change their signs Linear Equations II Example A. Solve. a. –5 = x + 6 –6 –5 = x –11 = x move +6 to the left, it turns into –6
  • 20. b. 3(x + 2) – x = 2(–2 + x) + x 3x + 6 – x = –4 + 2x + x 2x + 6 = –4 + 3x 4 + 6 = 3x – 2x 10 = x simplify each side move 2x and –4 to the other sides, change their signs Linear Equations II Example A. Solve. a. –5 = x + 6 –6 –5 = x –11 = x move +6 to the left, it turns into –6
  • 21. b. 3(x + 2) – x = 2(–2 + x) + x 3x + 6 – x = –4 + 2x + x 2x + 6 = –4 + 3x 4 + 6 = 3x – 2x 10 = x simplify each side move 2x and –4 to the other sides, change their signs Linear Equations II c. + x = 4 3 2 1 Example A. Solve. a. –5 = x + 6 –6 –5 = x –11 = x move +6 to the left, it turns into –6
  • 22. b. 3(x + 2) – x = 2(–2 + x) + x 3x + 6 – x = –4 + 2x + x 2x + 6 = –4 + 3x 4 + 6 = 3x – 2x 10 = x simplify each side move 2x and –4 to the other sides, change their signs Linear Equations II c. + x = 4 3 move to the other side, change its sign 3 42 1 Example A. Solve. a. –5 = x + 6 –6 –5 = x –11 = x move +6 to the left, it turns into –6
  • 23. b. 3(x + 2) – x = 2(–2 + x) + x 3x + 6 – x = –4 + 2x + x 2x + 6 = –4 + 3x 4 + 6 = 3x – 2x 10 = x simplify each side move 2x and –4 to the other sides, change their signs Linear Equations II c. + x = 4 3 move to the other side, change its sign 3 42 1 x = 2 1 4 3– Example A. Solve. a. –5 = x + 6 –6 –5 = x –11 = x move +6 to the left, it turns into –6
  • 24. b. 3(x + 2) – x = 2(–2 + x) + x 3x + 6 – x = –4 + 2x + x 2x + 6 = –4 + 3x 4 + 6 = 3x – 2x 10 = x simplify each side move 2x and –4 to the other sides, change their signs Linear Equations II c. + x = 4 3 move to the other side, change its sign 3 42 1 x = 2 1 4 3 x = – 4 2 4 3– Example A. Solve. a. –5 = x + 6 –6 –5 = x –11 = x move +6 to the left, it turns into –6
  • 25. b. 3(x + 2) – x = 2(–2 + x) + x 3x + 6 – x = –4 + 2x + x 2x + 6 = –4 + 3x 4 + 6 = 3x – 2x 10 = x simplify each side move 2x and –4 to the other sides, change their signs Linear Equations II c. + x = 4 3 move to the other side, change its sign 3 42 1 x = 2 1 4 3 x = x = 4 –1 – 4 2 4 3– Example A. Solve. a. –5 = x + 6 –6 –5 = x –11 = x move +6 to the left, it turns into –6
  • 26. II. (The Opposite Rule) If –x = c, then x = –c Linear Equations II
  • 27. II. (The Opposite Rule) If –x = c, then x = –c Example B. a. If –x = –15 Linear Equations II
  • 28. Linear Equations II II. (The Opposite Rule) If –x = c, then x = –c Example B. a. If –x = –15 then x = 15
  • 29. 4 3 Linear Equations II II. (The Opposite Rule) If –x = c, then x = –c Example B. a. If –x = –15 then x = 15 b. If –x =
  • 30. 4 3 4 3 Linear Equations II II. (The Opposite Rule) If –x = c, then x = –c Example B. a. If –x = –15 then x = 15 b. If –x = then x = –
  • 31. 4 3 4 3 Linear Equations II Recall that an equation or a relation expressed using fractions may always be restated in a way without using fractions. II. (The Opposite Rule) If –x = c, then x = –c Example B. a. If –x = –15 then x = 15 b. If –x = then x = –
  • 32. 4 3 4 3 Linear Equations II Recall that an equation or a relation expressed using fractions may always be restated in a way without using fractions. Example C. The value of 2/3 of an apple is the same as the value of ¾ of an orange, restate this relation in whole numbers. II. (The Opposite Rule) If –x = c, then x = –c Example B. a. If –x = –15 then x = 15 b. If –x = then x = –
  • 33. 4 3 4 3 Linear Equations II Recall that an equation or a relation expressed using fractions may always be restated in a way without using fractions. Example C. The value of 2/3 of an apple is the same as the value of ¾ of an orange, restate this relation in whole numbers. 2 3 A = 3 4 G,We've II. (The Opposite Rule) If –x = c, then x = –c Example B. a. If –x = –15 then x = 15 b. If –x = then x = –
  • 34. 4 3 4 3 Linear Equations II Recall that an equation or a relation expressed using fractions may always be restated in a way without using fractions. Example C. The value of 2/3 of an apple is the same as the value of ¾ of an orange, restate this relation in whole numbers. 2 3 A = 3 4 G,We've LCD =12, multiply 12 to both sides, II. (The Opposite Rule) If –x = c, then x = –c Example B. a. If –x = –15 then x = 15 b. If –x = then x = –
  • 35. 4 3 4 3 Linear Equations II Recall that an equation or a relation expressed using fractions may always be restated in a way without using fractions. Example C. The value of 2/3 of an apple is the same as the value of ¾ of an orange, restate this relation in whole numbers. 2 3 A = 3 4 G, ( ) *12 We've LCD =12, multiply 12 to both sides, 2 3 A = 3 4 G II. (The Opposite Rule) If –x = c, then x = –c Example B. a. If –x = –15 then x = 15 b. If –x = then x = –
  • 36. 4 3 4 3 Linear Equations II Recall that an equation or a relation expressed using fractions may always be restated in a way without using fractions. Example C. The value of 2/3 of an apple is the same as the value of ¾ of an orange, restate this relation in whole numbers. 2 3 A = 3 4 G, 4 ( ) *12 We've 2 3 A = 3 4 G LCD =12, multiply 12 to both sides, II. (The Opposite Rule) If –x = c, then x = –c Example B. a. If –x = –15 then x = 15 b. If –x = then x = –
  • 37. 4 3 4 3 Linear Equations II Recall that an equation or a relation expressed using fractions may always be restated in a way without using fractions. Example C. The value of 2/3 of an apple is the same as the value of ¾ of an orange, restate this relation in whole numbers. 2 3 A = 3 4 G, 34 ( ) *12 We've 2 3 A = 3 4 G LCD =12, multiply 12 to both sides, II. (The Opposite Rule) If –x = c, then x = –c Example B. a. If –x = –15 then x = 15 b. If –x = then x = –
  • 38. 4 3 4 3 Linear Equations II Recall that an equation or a relation expressed using fractions may always be restated in a way without using fractions. Example C. The value of 2/3 of an apple is the same as the value of ¾ of an orange, restate this relation in whole numbers. 2 3 A = 3 4 G, 34 8A = 9G ( ) *12 We've 2 3 A = 3 4 G or the value of 8 apples equal to 9 oranges. LCD =12, multiply 12 to both sides, II. (The Opposite Rule) If –x = c, then x = –c Example B. a. If –x = –15 then x = 15 b. If –x = then x = –
  • 39. Linear Equations II III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
  • 40. Example D. Solve the equations Linear Equations II x = –6 4 3a. III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
  • 41. III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve. Example D. Solve the equations Linear Equations II x = –6 4 3 x = –6 4 3 )( 4 a. clear the denominator, multiply both sides by 4
  • 42. Example D. Solve the equations Linear Equations II x = –6 4 3 x = –6 4 3 3x = –24 )( 4 a. clear the denominator, multiply both sides by 4 III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
  • 43. Example D. Solve the equations Linear Equations II x = –6 4 3 x = –6 4 3 3x = –24 x = –8 )( 4 a. clear the denominator, multiply both sides by 4 div by 3 III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
  • 44. Example D. Solve the equations x 4 3 1 2 6 5 4 3 Linear Equations II –+ = xb. x = –6 4 3 x = –6 4 3 3x = –24 x = –8 )( 4 a. clear the denominator, multiply both sides by 4 div by 3 III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
  • 45. Example D. Solve the equations x 4 3 1 2 6 5 4 3 the LCD = 12, multiply it to both sides to clear the denominators ( ) *12 Linear Equations II –+ = x x 4 3 1 2 6 5 4 3–+ = x b. x = –6 4 3 x = –6 4 3 3x = –24 x = –8 )( 4 a. clear the denominator, multiply both sides by 4 div by 3 III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
  • 46. Example D. Solve the equations x 4 3 1 2 6 5 4 3 the LCD = 12, multiply it to both sides to clear the denominators ( ) *12 3 Linear Equations II –+ = x x 4 3 1 2 6 5 4 3–+ = x b. x = –6 4 3 x = –6 4 3 3x = –24 x = –8 )( 4 a. clear the denominator, multiply both sides by 4 div by 3 III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
  • 47. Example D. Solve the equations x 4 3 1 2 6 5 4 3 the LCD = 12, multiply it to both sides to clear the denominators ( ) *12 3 4 Linear Equations II –+ = x x 4 3 1 2 6 5 4 3–+ = x b. x = –6 4 3 x = –6 4 3 3x = –24 x = –8 )( 4 a. clear the denominator, multiply both sides by 4 div by 3 III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
  • 48. Example D. Solve the equations x 4 3 1 2 6 5 4 3 the LCD = 12, multiply it to both sides to clear the denominators ( ) *12 3 4 2 3 Linear Equations II –+ = x x 4 3 1 2 6 5 4 3–+ = x b. x = –6 4 3 x = –6 4 3 3x = –24 x = –8 )( 4 a. clear the denominator, multiply both sides by 4 div by 3 III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
  • 49. Example D. Solve the equations x 4 3 1 2 6 5 4 3 the LCD = 12, multiply it to both sides to clear the denominators ( ) *12 3 4 2 3 3x + 8 = 10x – 9 Linear Equations II –+ = x x 4 3 1 2 6 5 4 3–+ = x b. x = –6 4 3 x = –6 4 3 3x = –24 x = –8 )( 4 a. clear the denominator, multiply both sides by 4 div by 3 III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
  • 50. Example D. Solve the equations x 4 3 1 2 6 5 4 3 the LCD = 12, multiply it to both sides to clear the denominators ( ) *12 3 4 2 3 3x + 8 = 10x – 9 Linear Equations II –+ = x x 4 3 1 2 6 5 4 3–+ = x b. x = –6 4 3 x = –6 4 3 3x = –24 x = –8 )( 4 a. clear the denominator, multiply both sides by 4 move 3x to the right and –9 to the left and switch signs. div by 3 III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
  • 51. Example D. Solve the equations x 4 3 1 2 6 5 4 3 the LCD = 12, multiply it to both sides to clear the denominators ( ) *12 3 4 2 3 3x + 8 = 10x – 9 9 + 8 = 10x – 3x Linear Equations II –+ = x x 4 3 1 2 6 5 4 3–+ = x b. x = –6 4 3 x = –6 4 3 3x = –24 x = –8 )( 4 a. clear the denominator, multiply both sides by 4 move 3x to the right and –9 to the left and switch signs. div by 3 III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
  • 52. Example D. Solve the equations x 4 3 1 2 6 5 4 3 the LCD = 12, multiply it to both sides to clear the denominators ( ) *12 3 4 2 3 3x + 8 = 10x – 9 9 + 8 = 10x – 3x 17 = 7x Linear Equations II –+ = x x 4 3 1 2 6 5 4 3–+ = x b. x = –6 4 3 x = –6 4 3 3x = –24 x = –8 )( 4 a. clear the denominator, multiply both sides by 4 move 3x to the right and –9 to the left and switch signs. div by 3 III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
  • 53. Example D. Solve the equations x 4 3 1 2 6 5 4 3 the LCD = 12, multiply it to both sides to clear the denominators ( ) *12 3 4 2 3 3x + 8 = 10x – 9 9 + 8 = 10x – 3x 17 = 7x = x 7 17 Linear Equations II –+ = x x 4 3 1 2 6 5 4 3–+ = x b. x = –6 4 3 x = –6 4 3 3x = –24 x = –8 )( 4 a. clear the denominator, multiply both sides by 4 move 3x to the right and –9 to the left and switch signs. div by 3 div by 7 III. (Fractional Equations Rule) Multiply fractional equations by the LCD to remove the fractions first, then solve.
  • 54. (x – 20) = x – 27 100 15 100 45 Linear Equations II c.
  • 55. (x – 20) = x – 27 100 15 100 45 multiply 100 to both sides to remove denominators [ (x – 20) = x – 27 100 15 100 45 ] * 100 Linear Equations II c.
  • 56. (x – 20) = x – 27 100 15 100 45 multiply 100 to both sides to remove denominators [ (x – 20) = x – 27 100 15 100 45 ] * 100 1 1 100 Linear Equations II c.
  • 57. (x – 20) = x – 27 100 15 100 45 multiply 100 to both sides to remove denominators [ (x – 20) = x – 27 100 15 100 45 ] * 100 1 1 100 15 (x – 20) = 45x – 2700 Linear Equations II c.
  • 58. (x – 20) = x – 27 100 15 100 45 multiply 100 to both sides to remove denominators [ (x – 20) = x – 27 100 15 100 45 ] * 100 1 1 100 15 (x – 20) = 45x – 2700 15x – 300 = 45x – 2700 Linear Equations II c.
  • 59. (x – 20) = x – 27 100 15 100 45 multiply 100 to both sides to remove denominators [ (x – 20) = x – 27 100 15 100 45 ] * 100 1 1 100 15 (x – 20) = 45x – 2700 15x – 300 = 45x – 2700 2700 – 300 = 45x –15x Linear Equations II c.
  • 60. (x – 20) = x – 27 100 15 100 45 multiply 100 to both sides to remove denominators [ (x – 20) = x – 27 100 15 100 45 ] * 100 1 1 100 15 (x – 20) = 45x – 2700 15x – 300 = 45x – 2700 2700 – 300 = 45x –15x 2400 = 30x Linear Equations II c.
  • 61. (x – 20) = x – 27 100 15 100 45 multiply 100 to both sides to remove denominators [ (x – 20) = x – 27 100 15 100 45 ] * 100 1 1 100 15 (x – 20) = 45x – 2700 15x – 300 = 45x – 2700 2700 – 300 = 45x –15x 2400 = 30x 2400 / 30 = x 80 = x Linear Equations II c.
  • 62. (x – 20) = x – 27 100 15 100 45 multiply 100 to both sides to remove denominators [ (x – 20) = x – 27 100 15 100 45 ] * 100 1 1 100 15 (x – 20) = 45x – 2700 15x – 300 = 45x – 2700 2700 – 300 = 45x –15x 2400 = 30x 2400 / 30 = x 80 = x Linear Equations II c. d. 0.25(x – 100) = 0.10x – 1
  • 63. (x – 20) = x – 27 100 15 100 45 multiply 100 to both sides to remove denominators [ (x – 20) = x – 27 100 15 100 45 ] * 100 1 1 100 15 (x – 20) = 45x – 2700 15x – 300 = 45x – 2700 2700 – 300 = 45x –15x 2400 = 30x 2400 / 30 = x 80 = x Linear Equations II c. (x – 100) = x – 1100 25 100 10 d. 0.25(x – 100) = 0.10x – 1 Change the decimal into fractions, we get
  • 64. (x – 20) = x – 27 100 15 100 45 multiply 100 to both sides to remove denominators [ (x – 20) = x – 27 100 15 100 45 ] * 100 1 1 100 15 (x – 20) = 45x – 2700 15x – 300 = 45x – 2700 2700 – 300 = 45x –15x 2400 = 30x 2400 / 30 = x 80 = x Linear Equations II c. (x – 100) = x – 1100 25 100 10 multiply 100 to both sides to remove denominators d. 0.25(x – 100) = 0.10x – 1 Change the decimal into fractions, we get You finish it…
  • 65. (x – 20) = x – 27 100 15 100 45 multiply 100 to both sides to remove denominators [ (x – 20) = x – 27 100 15 100 45 ] * 100 1 1 100 15 (x – 20) = 45x – 2700 15x – 300 = 45x – 2700 2700 – 300 = 45x –15x 2400 = 30x 2400 / 30 = x 80 = x Linear Equations II c. (x – 100) = x – 1100 25 100 10 multiply 100 to both sides to remove denominators d. 0.25(x – 100) = 0.10x – 1 Change the decimal into fractions, we get You finish it… Ans: x =160
  • 66. Linear Equations II IV. (Reduction Rule) Use division to reduce equations to simpler ones.
  • 67. Linear Equations II IV. (Reduction Rule) Use division to reduce equations to simpler ones. Specifically divide the common factor of the coefficients of each term to make them smaller and easier to work with.
  • 68. 14x – 49 = 70x – 98 Linear Equations II Example. E. Simplify the equation first then solve. IV. (Reduction Rule) Use division to reduce equations to simpler ones. Specifically divide the common factor of the coefficients of each term to make them smaller and easier to work with.
  • 69. 14x – 49 = 70x – 98 divide each term by 7 Linear Equations II Example. E. Simplify the equation first then solve. 14x – 49 70x – 98= 7 7 7 7 IV. (Reduction Rule) Use division to reduce equations to simpler ones. Specifically divide the common factor of the coefficients of each term to make them smaller and easier to work with.
  • 70. 14x – 49 = 70x – 98 divide each term by 7 Linear Equations II Example. E. Simplify the equation first then solve. 14x – 49 70x – 98= 7 7 7 7 2x – 7 = 10x – 14 IV. (Reduction Rule) Use division to reduce equations to simpler ones. Specifically divide the common factor of the coefficients of each term to make them smaller and easier to work with.
  • 71. 14x – 49 = 70x – 98 divide each term by 7 Linear Equations II Example. E. Simplify the equation first then solve. 14x – 49 70x – 98= 7 7 7 7 2x – 7 = 10x – 14 14 – 7 = 10x – 2x IV. (Reduction Rule) Use division to reduce equations to simpler ones. Specifically divide the common factor of the coefficients of each term to make them smaller and easier to work with.
  • 72. 14x – 49 = 70x – 98 divide each term by 7 Linear Equations II Example. E. Simplify the equation first then solve. 14x – 49 70x – 98= 7 7 7 7 2x – 7 = 10x – 14 14 – 7 = 10x – 2x 7 = 8x = x 8 7 IV. (Reduction Rule) Use division to reduce equations to simpler ones. Specifically divide the common factor of the coefficients of each term to make them smaller and easier to work with.
  • 73. 14x – 49 = 70x – 98 divide each term by 7 Linear Equations II Example. E. Simplify the equation first then solve. 14x – 49 70x – 98= 7 7 7 7 2x – 7 = 10x – 14 14 – 7 = 10x – 2x 7 = 8x = x 8 7 We should always reduce the equation first if it is possible. IV. (Reduction Rule) Use division to reduce equations to simpler ones. Specifically divide the common factor of the coefficients of each term to make them smaller and easier to work with.
  • 74. There’re two type of equations that give unusual results. The first type is referred to as identities. Linear Equations II
  • 75. There’re two type of equations that give unusual results. The first type is referred to as identities. A simple identity is the equation “x = x”. Linear Equations II
  • 76. There’re two type of equations that give unusual results. The first type is referred to as identities. A simple identity is the equation “x = x”. This corresponds to the trick question “What number x is equal to itself?” Linear Equations II
  • 77. There’re two type of equations that give unusual results. The first type is referred to as identities. A simple identity is the equation “x = x”. This corresponds to the trick question “What number x is equal to itself?” The answer of course is that x can be any number or that the solutions of equation “x = x” are all numbers. Linear Equations II
  • 78. There’re two type of equations that give unusual results. The first type is referred to as identities. A simple identity is the equation “x = x”. This corresponds to the trick question “What number x is equal to itself?” The answer of course is that x can be any number or that the solutions of equation “x = x” are all numbers. Linear Equations II This is also the case for the any equation where both sides are identical such as 2x + 1 = 2x + 1,1 – 4x = 1 – 4x etc…
  • 79. There’re two type of equations that give unusual results. The first type is referred to as identities. A simple identity is the equation “x = x”. This corresponds to the trick question “What number x is equal to itself?” The answer of course is that x can be any number or that the solutions of equation “x = x” are all numbers. Linear Equations II This is also the case for the any equation where both sides are identical such as 2x + 1 = 2x + 1,1 – 4x = 1 – 4x etc… An equation with identical expressions on both sides or can be rearranged into identical sides has all numbers as its solutions. Such an equation is called an identity.
  • 80. There’re two type of equations that give unusual results. The first type is referred to as identities. A simple identity is the equation “x = x”. This corresponds to the trick question “What number x is equal to itself?” The answer of course is that x can be any number or that the solutions of equation “x = x” are all numbers. Example F. Solve. Linear Equations II 2(x – 1) + 3 = x – (– x –1) This is also the case for the any equation where both sides are identical such as 2x + 1 = 2x + 1,1 – 4x = 1 – 4x etc… An equation with identical expressions on both sides or can be rearranged into identical sides has all numbers as its solutions. Such an equation is called an identity.
  • 81. There’re two type of equations that give unusual results. The first type is referred to as identities. A simple identity is the equation “x = x”. This corresponds to the trick question “What number x is equal to itself?” The answer of course is that x can be any number or that the solutions of equation “x = x” are all numbers. Example F. Solve. Linear Equations II 2(x – 1) + 3 = x – (– x –1) This is also the case for the any equation where both sides are identical such as 2x + 1 = 2x + 1,1 – 4x = 1 – 4x etc… expand 2x – 2 + 3 = x + x + 1 An equation with identical expressions on both sides or can be rearranged into identical sides has all numbers as its solutions. Such an equation is called an identity.
  • 82. There’re two type of equations that give unusual results. The first type is referred to as identities. A simple identity is the equation “x = x”. This corresponds to the trick question “What number x is equal to itself?” The answer of course is that x can be any number or that the solutions of equation “x = x” are all numbers. Example F. Solve. Linear Equations II 2(x – 1) + 3 = x – (– x –1) This is also the case for the any equation where both sides are identical such as 2x + 1 = 2x + 1,1 – 4x = 1 – 4x etc… expand 2x – 2 + 3 = x + x + 1 simplify 2x + 1 = 2x + 1 An equation with identical expressions on both sides or can be rearranged into identical sides has all numbers as its solutions. Such an equation is called an identity.
  • 83. There’re two type of equations that give unusual results. The first type is referred to as identities. A simple identity is the equation “x = x”. This corresponds to the trick question “What number x is equal to itself?” The answer of course is that x can be any number or that the solutions of equation “x = x” are all numbers. Example F. Solve. Linear Equations II 2(x – 1) + 3 = x – (– x –1) This is also the case for the any equation where both sides are identical such as 2x + 1 = 2x + 1,1 – 4x = 1 – 4x etc… expand 2x – 2 + 3 = x + x + 1 simplify 2x + 1 = 2x + 1 An equation with identical expressions on both sides or can be rearranged into identical sides has all numbers as its solutions. Such an equation is called an identity. two sides are identical
  • 84. There’re two type of equations that give unusual results. The first type is referred to as identities. A simple identity is the equation “x = x”. This corresponds to the trick question “What number x is equal to itself?” The answer of course is that x can be any number or that the solutions of equation “x = x” are all numbers. Example F. Solve. Linear Equations II 2(x – 1) + 3 = x – (– x –1) This is also the case for the any equation where both sides are identical such as 2x + 1 = 2x + 1,1 – 4x = 1 – 4x etc… expand 2x – 2 + 3 = x + x + 1 simplify 2x + 1 = 2x + 1 So this equation is an identity and every number is a solution. An equation with identical expressions on both sides or can be rearranged into identical sides has all numbers as its solutions. Such an equation is called an identity. two sides are identical
  • 85. At the opposite end of the identities are the “impossible” equations where there is no solution at all. Linear Equations II
  • 86. At the opposite end of the identities are the “impossible” equations where there is no solution at all. An example is the equation x = x + 1. Linear Equations II
  • 87. At the opposite end of the identities are the “impossible” equations where there is no solution at all. An example is the equation x = x + 1. This corresponds to the trick question “What number is still the same after we add 1 to it?” Linear Equations II
  • 88. At the opposite end of the identities are the “impossible” equations where there is no solution at all. An example is the equation x = x + 1. This corresponds to the trick question “What number is still the same after we add 1 to it?” Of course there no such number. Linear Equations II
  • 89. At the opposite end of the identities are the “impossible” equations where there is no solution at all. An example is the equation x = x + 1. This corresponds to the trick question “What number is still the same after we add 1 to it?” Of course there no such number. Linear Equations II If we attempt to solve x = x + 1 x – x = 1
  • 90. At the opposite end of the identities are the “impossible” equations where there is no solution at all. An example is the equation x = x + 1. This corresponds to the trick question “What number is still the same after we add 1 to it?” Of course there no such number. Linear Equations II If we attempt to solve x = x + 1 x – x = 1 0 = 1we get which is an impossibility.
  • 91. At the opposite end of the identities are the “impossible” equations where there is no solution at all. An example is the equation x = x + 1. This corresponds to the trick question “What number is still the same after we add 1 to it?” Of course there no such number. Linear Equations II If we attempt to solve x = x + 1 x – x = 1 0 = 1we get which is an impossibility. These equations are called inconsistent equations.
  • 92. At the opposite end of the identities are the “impossible” equations where there is no solution at all. An example is the equation x = x + 1. This corresponds to the trick question “What number is still the same after we add 1 to it?” Of course there no such number. Example F. Solve the equation Linear Equations II 2(x – 1) + 4 = x – (– x –1) If we attempt to solve x = x + 1 x – x = 1 0 = 1we get which is an impossibility. These equations are called inconsistent equations.
  • 93. At the opposite end of the identities are the “impossible” equations where there is no solution at all. An example is the equation x = x + 1. This corresponds to the trick question “What number is still the same after we add 1 to it?” Of course there no such number. Example F. Solve the equation Linear Equations II 2(x – 1) + 4 = x – (– x –1) If we attempt to solve x = x + 1 x – x = 1 0 = 1we get expand 2x – 2 + 4 = x + x + 1 which is an impossibility. These equations are called inconsistent equations.
  • 94. At the opposite end of the identities are the “impossible” equations where there is no solution at all. An example is the equation x = x + 1. This corresponds to the trick question “What number is still the same after we add 1 to it?” Of course there no such number. Example F. Solve the equation Linear Equations II 2(x – 1) + 4 = x – (– x –1) If we attempt to solve x = x + 1 x – x = 1 0 = 1we get expand 2x – 2 + 4 = x + x + 1 simplify 2x + 2 = 2x + 1 which is an impossibility. These equations are called inconsistent equations.
  • 95. At the opposite end of the identities are the “impossible” equations where there is no solution at all. An example is the equation x = x + 1. This corresponds to the trick question “What number is still the same after we add 1 to it?” Of course there no such number. Example F. Solve the equation Linear Equations II 2(x – 1) + 4 = x – (– x –1) If we attempt to solve x = x + 1 x – x = 1 0 = 1we get expand 2x – 2 + 4 = x + x + 1 simplify 2x + 2 = 2x + 1 2 = 1 So this is an inconsistent equation and there is no solution. which is an impossibility. These equations are called inconsistent equations.
  • 96. Exercise. A. Solve for x using the switch-side-switch-sign rule. Remember to move the x’s first and get positive x’s. 1. x + 2 = 5 – 2x 2. 2x – 1 = – x – 7 3. –x = x – 8 4. –x = 3 – 2x 5. –5x = 6 – 3x 6. –x + 2 = 3 + 2x 7. –3x – 1= 3 – 6x 8. –x + 7 = 3 – 3x 9. –2x + 2 = 9 + x Linear Equations II x = 5 2x = – 7 = x B. Solve the following fractional equations by using the LCD to remove the denominators first. x = –2 5 3 10. x = –5 3 –4 11. = 4 3x 12. 2 –1 = 3 9x 13. 2 3 x = 3 –2 14. 2 –1 = 6 7x 15. 4 –3
  • 97. Linear Equations II x 6 3 1 2 3 5 2 3–+ = x16. x 4 6 –3 1 8 –5 – 1– = x17. x 4 5 3 2 10 7 4 3+– = x18. x 8 12 –5 7 16 –5 + 1+ = x19. (x – 20) = x – 3 100 30 100 2020. (x + 5) – 3 = (x – 5) 100 25 100 2021. (x +15) = x + 1 100 15 100 3522. (50 – x) + 2 = (x – 50) 100 25 100 2023. C. Reduce the equations then solve. 28. –3x – 12 = 30 – 6x 29. 15x – 10x = 25x – 20 30. –4(x – 3) = 12(x + 2) – 8x 31. 15x – 10(x + 2) = 25x – 20 24. –0.3x – 0.25 = 1 – 0.6x 25. 0.15x – 0.1x = 0.25x – 2.4 26. 0.37 – 0.17x = 0.19x – 0.1 27. 1.7x – 0.11 = 0.22 – 0.4x
  • 98. Linear Equations II D. Identify which equations are identities and which are inconsistent. 34. –x + 1 = 5x – 2(3x + 1) 35. –2(x + 2) = 5x – (7x – 4) 36. 4(x – 3) – 2 = 1 – (14 – 4x) 37. 4(x – 3) – 2 = 1 – (15 – 4x) 32. –x + 1 = –x + 2 33. x + 2 = 5 – (3 – x)