1. Objective
Solve equations in one variable that contain variable terms on both sides.

2. Example 1: Solving Equations with Variables on Both Sides
Solve 5 - n – 2 = 6.
3 – n = 6
To collect the variable terms on one side, subtract 2 from 5. –
-n = 3
3 is hanging on, so subtract 3 from both sides.
Since -n is negative, divide both sides by -1.
n = -3

3. Example 1: Solving Equations with Variables on Both Sides
Solve 7n – 2 = 5n + 6.
7n – 2 = 5n + 6
To collect the variable terms on one side, subtract 5n from both sides.
–5n –5n
2n – 2 =
2n =
Since n is multiplied by 2, divide both sides by 2 to undo the multiplication.
n = 4

4. Check It Out! Example 1a
Solve 4b + 2 = 3b.
4b + 2 = 3b
To collect the variable terms on one side, subtract 3b from both sides.
-3b –3b
b + 2 = 0
– 2 – 2
b = –2

5. Check It Out! Example 1b
Solve 5 - 3y = -7y + 3.
To collect the variable terms on one side, add 4y to both sides.
5 - 3y = -7y + 3
+ 7y + 7y
5 + 4y = 3
Since 5 is added to 4y, subtract 5 from both sides to undo the addition.
4y = -2
Since y is multiplied by 4, divide both sides by 4 to undo the multiplication.
y = -½

6. Check It Out! Example 1b
Solve 2 + 4y = 9y – 8.
To collect the variable terms on one side, subtract 4y from both sides.
2 + 4y = 9y – 8
–4y –4y
= 5y – 8
Since 8 is subtracted from 5y, add 8 to both sides to undo the subtraction.
= 5y
Since y is multiplied by 5, divide both sides by 5 to undo the multiplication.
2 = y

7. Check It Out! Example 2b
Solve -2x = 2(x + 2).
-2x = 2(x + 2)
Distribute 2 to the expression in parentheses.
-2x = 2(x) + 2(2)
-2x = 2x + 4
-2x + 16 = 2x + 4
Combine like terms.
+2x x
To collect the variable terms on one side, add 2x to both sides.
16 = 4x + 4
– – 4
Subtract 4 from both sides
Divide by 4 and simplify.
12 = 4x
3 = x

8. Example 2: Simplifying Each Side Before Solving Equations
Solve 12 – 6a + 4a = –1 – 5(7 – 2a).
12 – 6a + 4a = –1 –5(7 – 2a)
Distribute –5 to the expression in parentheses.
12 – 6a + 4a = –1 –5(7) –5(–2a)
12 – 6a + 4a = –1 – a
12 – 2a = – a
Combine like terms.
To collect the variable terms on one side, add 2a to both sides.
Since –36 is added to 10a, add 36 to both sides.
+ 2a a
= a
= a

9. Example 2 Continued
Solve 4 – 6a + 4a = –1 – 5(7 – 2a).
48 = 12a
Since a is multiplied by 12, divide both sides by 12.
4 = a