1. 2-3 Equations With Variables on Both Sides
Hubarth
Algebra

2. Ex 1 Variables on Both Sides
To solve an equation that has variables on both sides, use the Addition or Subtraction
Properties of Equality to get the variables on one side of the equation.
With these problems the first thing is to get all variables on one side of the equation
Ex 1 Variables on Both Sides
Solve 5x – 3 = 2x + 12.
5x – 3 – 2x = 2x + 12 – 2x Subtract 2x from each side.
3x – 3 = 12 Combine like terms.
3x – = Add 3 to each side.
3x = 15 Simplify.
= Divide each side by 3.
3x 3
15 3
x = 5 Simplify.

3. Special Cases: Identities and No Solutions
An equation has no solution if no variable makes the equation true.
Example 2x = 2x + 1
-2x -2x
0 = 1
This is never true, therefore no solution.
An equation that is true for every value of the variable is an identity.
Example 3x = 3x
Any time you have the same value equal to itself, you will have an identity or infinite
number of solution (many solutions).

4. Ex 2 Identities and Equations With No Solution
Solve each equation. If the equation is an identity, write identity. If it has no solution write
no solution.
4 – 4y = –2(2y – 2)
4 – 4y = –4y Use the Distributive Property.
4 – 4y + 4y = –4y y Add 4y to each side.
4 = Always true!
The equation is true for every value of y, so the equation is an identity.

5. Ex 2 Identities and Equations With No Solution
Solve each equation. If the equation is an identity, write identity. If it has no solution write
no solution.
–6z + 8 = z + 10 – 7z
–6z + 8 = –6z Combine like terms.
–6z z = –6z z Add 6z to each side.
8 = Not true for any value of z!
This equation has no solution.

6. Practice
Solve each equation.
-6d = d b. 2(c – 6) = 9c + 2
c. m – 5 = 3m d. 7k – 4 = 5k + 16
d =
c = - 2
m =
k = 10
2. Determine whether the equation is an identity or whether it has no solution.
a n = 5n – 1 b. 10 – 8a = 2(5 – 4a)
No solution
identity