Download Solving Equations with Variables on Both Sides

Adapted by Mrs. Garay
Warm Up
Solve.
1. 2x + 9x – 3x + 8 = 16
2. –4 = 6x + 22 – 4x
x = –13
1
3. 2 +x = 57 x = 34
7 7
4. 9x – 2x = 3
16 4
1
8
x=1
x = 50
Learn to solve equations with variables on
both sides of the equal sign.
Some problems produce equations that have
variables on both sides of the equal sign.
Solving an equation with variables on both
sides is similar to solving an equation with a
variable on only one side. You can add or
subtract a term containing a variable on both
sides of an equation.
Example 1A: Solving Equations with Variables on Both Sides
Solve.
4x + 6 = x
4x + 6 = x
– 4x
– 4x
6 = –3x
6 = –3x
–3
–3
–2 = x
Subtract 4x from both sides.
Divide both sides by –3.
Helpful Hint
Check your solution by substituting the value
back into the original equation. For example,
4(-2) + 6 = -2 or -2 = -2.
Example 1B: Solving Equations with Variables on Both Sides
Solve.
9b – 6 = 5b + 18
9b – 6 = 5b + 18
– 5b
– 5b
Subtract 5b from both sides.
4b – 6 = 18
+6 +6
4b = 24
4b = 24
4
4
b=6
Add 6 to both sides.
Divide both sides by 4.
Example 1C: Solving Equations with Variables on Both Sides
Solve.
9w + 3 = 9w + 7
9w + 3 = 9w + 7
– 9w
– 9w
3≠
Subtract 9w from both sides.
7
No solution. There is no number that can be
substituted for the variable w to make the
equation true.
Helpful Hint
If the variables in an equation are eliminated
and the resulting statement is false, the
equation has no solution.
Your Turn!
Solve.
5x + 8 = x
5x + 8 = x
– 5x
– 5x
8 = –4x
8 = –4x
–4
–4
–2 = x
Subtract 5x from both sides.
Divide both sides by –4.
Your Turn Again!
Solve.
3b – 2 = 2b + 12
3b – 2 = 2b + 12
– 2b
– 2b
Subtract 2b from both sides.
b–2=
+2
b
=
12
+ 2 Add 2 to both sides.
14
One more!!!!!!!!
Solve.
3w + 1 = 3w + 8
3w + 1 = 3w + 8
– 3w
– 3w
1≠
Subtract 3w from both sides.
8
No solution. There is no number that can be
substituted for the variable w to make the
equation true.
To solve multi-step equations with variables on
both sides, first combine like terms and clear
fractions. Then add or subtract variable terms
to both sides so that the variable occurs on
only one side of the equation. Then use
properties of equality to isolate the variable.
Example 2: Solving Multi-Step Equations with Variables on Both
Sides
Solve.
10z – 15 – 4z = 8 – 2z - 15
10z – 15 – 4z = 8 – 2z – 15
6z – 15 = –2z – 7 Combine like terms.
+ 2z
+ 2z
Add 2z to both sides.
8z – 15
+ 15
8z
8z
8
z
=
=8
= 8
8
=1
–7
+15 Add 15 to both sides.
Divide both sides by 8.
Your Turn!
Solve.
12z – 12 – 4z = 6 – 2z + 32
12z – 12 – 4z = 6 – 2z + 32
8z – 12 = –2z + 38 Combine like terms.
+ 2z
+ 2z
Add 2z to both sides.
10z – 12 =
38
+ 12
+12 Add 12 to both sides.
10z = 50
10z = 50
Divide both sides by 10.
10
10
z=5