Linear equations powerpoint

LINEAR EQUATIONS: AN
INTRODUCTION (G9)
DEFINITION AND EXAMPLES

DEFINITION
AN EQUATION: is a statement that says that 2
expressions are equal.
For example two R5 coins is equivalent to R10
we can write it down as 2(R5) = R10

What you do on the left should be done on
the right to keep the equation balanced
2(R5) = R10
2(R5) – R1 = R10 –R1
2(R5) + R20 = R10 + R20
X + 3 = 5

X + 3 = 5
x+3-3 = 5 -3
x = 2
(2) + 3 = 5

X + 6 = 15
x +6 -6 = 15 -6
x = 9
2x + 3 = 13

2x + 3 = 13
2x +3 -3 = 13 -3
2x = 10
2x/2 = 10/2
x = 5
2(x +4) =20

2(x +4) = 20
2x + 8 =20
2x +8 – 8 = 20 -8
2x = 12
2x/2 = 12/2
x = 6
2(x+4) = 20
2(x+4)/2 = 20/2
X + 4 = 10
X = 10-4
X= 6

2x -3 = 4x -6
2x – 4x = 3-6
-2x = -3
-2x/-2 = -3/-2
x= 3/2
2x +10 = 3x - 4

2x +10 = 3x – 4
2x -3x = -4-10
-x = -14
-x/-1 =-14/-1
x = 14
X squared -9 = 0
(x +3)(x -3) = 0
(x+3) = 0 or (x-3) = 0
X= -3 or x = 3

DIFFERENCE OF TWO SQUARES (DOTS)
X2 – 9 = 0
(x +3)(x-3) =0
x+3 = 0 or x – 3 = 0
x = 3 or x = -3

X2 – 25 = 0
(x+5)(x-5) = 0
x+5 = 0 or x- 5 = 0
x = -5 or x = 5
2x2 – 2 = 0
2(x2 -1) = 0 (divide both sides by 2)
x2 -1 = 0
(x+1)(x-1) =0
x+1 = 0 or x-1 =0
x= -1 or x = 1

25(x2 – 4) = 0 (divide both sides by 25)
x2 – 4 = 0
(x-2)(x+2) = 0
x-2 = 0 or x+2 =0
x = 2 or x = -2
X2 – 7 = 9
x2 – 7-9 =0
x2-16 = 0
(x-4)(x+4) = 0
x- 4 = 0 or x+4 =o
x= 4 or x = -4

X3 – x = 0
x(x2-1) =0
x(x+1)(x-1) = 0
x= 0 or x+1 = 0 or x-1 =0
x=0 or x= -1 or x= 1
the power of x tells us how many values
of x will make up our solution.

EXERCISE
 SOLVE FOR X IN THE FOLLOWING EQUATIONS
 1. x + 1 = 6 x = 5
 2. x – 6 = 1 x = 7
 3. 2x + 4 = 10 x = 3
 4. 3x – 6 = 14 -2x x = 4
 5. X2 – 81 = 0 x = 9 or x = -9
 6. X2 – 4 = 0 x = 2 or x = -2
 7. 2x2 – 32 = 0 x = 4 or x = -4
 8. 25x2 – 1 = 0 x = 1/5 or x = -1/5

TRINOMIAL EQUATIONS
 1. (x-2)(x-3) = 0 x2 -5x + 6 = 0
 x-2 =0 or x-3 = 0 (2x2)-5(2) +6 = 4-10+6
 X = 2 or x = 3 = -6+6 =0
 2. (x-4)(x+5) = 0 x2 + x -20 = 0
 X- 4 = 0 or x + 5 = 0 (x -4 )(x+5 ) = 0
 X = 4 or x= - 5

3. (x+2)(x+6) = 0 x2 + 8x + 12 = 0
x+2 = 0 or x+6 = 0
x= -2 or x = -6

(x-3)(x+4) = -6
x2 + x – 12 = -6
X2 + x -12 + 6 = 0
X2 +x – 6 = 0
(x +3)(x-2) = 0
X2 – 25 =0

EXERCISE
Solve for x in the following
1. x + 4 = 10 x = 6
2. x -15 = 3 x = 18
3. 2x + 5 = 3x- 10 x = 15
4. X2 = 25 x= 5 or x = -5
5. X2 +10 = 11 x = 1 or x = -1
6. 3x2 + 6 = 18 x = -2 or x = 2
7. (x+3)(x+6) = 0 x = -3 or x = -6
8. X2 + 4x – 12 = 0 x = -6 or x = 2
9. X2 + 5x = -6 x = -2 or x = -3

SOLUTIONS
1. x + 4 = 10
x = 10 – 4
x = 6
2. x -15 = 3
x = 3 + 15
x = 18
3. 2x + 5 = 3x- 10
5+10 = 3x -2x
15 = x
x = 15

X2 + 4x – 12 = 0
(x +6)(x-2) = 0
x = -6 or x = 2
X2 + 5x = -6
X2 + 5x +6 = 0
(x+2)(x+3) = 0
x = -2 or x = -3

EQUATIONS WITH FRACTIONS

1
2
𝑥 = 10
X = 10 x 2
X= 20

EQU…

3
5
𝑥 = 15
3x = 15 x 5
3x = 75
x = 75/3
x= 25

EQU…

4
5
𝑥 =
3
4
𝑥 − 4

4
5
𝑥 −
3
4
𝑥 = −4

4 4𝑥 −5 3𝑥
20
= −4
16x – 15x = - 4 x 20
x = - 80

CLASSWORK: solve for x
1.
1
3
𝑥 = 9 x = 27
2.
2
5
𝑥 = 10 x = 25
3.
7
8
𝑥 = 1 x = 8/7
4.
1
2
𝑥 +
1
3
𝑥 = 5 x = 6
5.
3
4
𝑥 − 4 = 0 x = 16/3
6.
3
7
𝑥 + 2 =
1
4
𝑥 − 3 x = -28
7. X2-
1
4
= 0 x = ½ or x= -1/2

EXPONENTIAL EQUATIONS
 EXAMPLES
1. 2X = 27
X = 7
2. 5X = 56
x= 6
3. X4 = 34
x = 3
2 is the base
7 is the exponent
When the basis are the same, then equate the
powers. If powers are the same, then equate
the bases.

Expo…
1. 2x3 = 54
x3 = 27
x3 = 33
x = 3
2. (3x)3 = 3
33x = 31
3x = 1
x = 1/3
Write 27 as a power of
3
Don’t multiply a base and a coefficient.
NOTES

Expo..
1. 2x – 25 = -17
2x = -17 +25
2x = 8
2x = 23
x= 3

CLASSWORK
1. 2x = 128 x = 7
2. 5x = 53 x = 3
3. 7x = 343 x = 3
4. 2x + 9 = 25 x = 4
5. 27(3x) = 3 x = -2
6. 7x – 9 = -8 x = 0
7. 8x-8 = 8 x = 9
8. 4x + 7 = 23 x = 2

Linear equations powerpoint

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Linear equations powerpoint