2. Solving One-Step Equations:
“An equation is like a balance scale
because it shows that two
quantities are equal. The scales
remained balanced when the same
weight is added (or removed from)
to each side.”
x + 2 = 5 x + (-5) = -2
3. What does it mean to Solve an equation?
“To solve an equation containing a
variable, you find the value (or values)
of the variable that make the equation
true.”
“Get the variable alone on one side of
the equal sign…using inverse
operations, which are operations that
undo each other.”
4. Inverse Operations
Addition and Subtraction are inverse
operations because they undo each
other.
Multiplication and division are inverse
operations because they undo each
other.
5. Using Reciprocals:
In order to solve the equation
above, you need to divide by 2/3.
Remember: To divide a fraction,
you multiply by its reciprocal. In
other words: flip it!
2/3x = 12
3/2*2/3x= 1x
3/2* 12/1= 18
So x = 18.
6. Solving Two-Step Equations
“A two-step equation is an equation
that involves two operations.”
PEMDAS tells us to multiply or divide
before we add or subtract, but to solve
equations, we do just the opposite: we
add or subtract before we multiply
or divide.
7. To Solve Multi-Step Equations:
1) “Clear the equation of
fractions and decimals.”
2) Apply the Distributive
Property as needed.
3) “Combine like terms.”
4) “Undo addition and
subtraction.”
5) “Undo multiplication and
division.”
8. Multi-step equations
• We have added to the level of difficulty by solving equations with 2
steps, by combining like terms first, and by using the distributive
property.
• Now we increase the difficulty again by solving equations with
fractions and decimals.
9. Equations with Variables on Both Sides:
Use the Addition or Subtraction
property of Equality to get the
variables on one side of the
equation.
10. Example
• For example: 2
6 22
3
n
Our steps are the same in this problem. First we
add six to both sides, then we multiply both sides
by the reciprocal 3/2 and then solve. Let’s try it.
12. Another example
• Sometimes we have to
distribute the fraction
like this:
2
( 6) 3
3
m
2
4 3
3
m
2
7
3
3 2 3
7
2 3 2
21 1
10
2 2
m
m
m or
15. Try This
• 1.2n + 3.4 = 10 n = 5.5
• (a + 6) = 2 a = -5/2
4
7
16. Another example
14
8 10
2
n
14
18
2
n
14 36
n
First add 8 to both sides
Next multiply both sides by 2
(2) (2)
Subtract 14 from both sides and solve.
+8 +8
-14 -14
N = 22
20. Clear the equation of fractions
•Multiply each side by the
LCD to get rid of the
fraction or fractions.
21. Solving linear equations involving fractions
3.)
4
5
8
2
3
x
x
To remove fractions from an equation:
Multiply both sides of the equation (each term) by the least common
denominator
Multiply both sides of the
equation by 10 (each term)
x = 40
22. 3 - 2x = 4x – 6
+ 2x +2x
3 = 6x – 6
+ 6 + 6
9 = 6x
6 6
or 1.5 = x
Solve
1. Draw “the river”
2. Clear the fraction – multiply
each term by the LCD
3. Simplify
4. Add 2x to both sides
5. Simplify
6. Add 6 to both sides
7. Simplify
8. Divide both sides by 6
9. Simplify
10. Check your answer
1.5
3 1 1 3
8 4 2 4
1.5
3
8
1
4
x
1
2
x
3
4
(8)
3
8
(8)
1
4
x (8)
1
2
x (8)
3
4
3
2
23. Special Case #1
6) 2x + 5 = 2x - 3
-2x -2x
5 = -3
This is never true!
No solutions
1. Draw “the river”
2. Subtract 2x from both
sides
3. Simplify
