Lesson 3: Solving Equations With a Variable on Both Sides

Property

We'll start by choosing a variable side and a constant side, and then use the Subtraction and Addition Properties of Equality to collect all variables on one side and all constants on the other side.
Remember, what you do to the left side of the equation, you must do to the right side too.

Examples

• In $6y = 5y + 10$, subtract $5y$ from both sides. This simplifies directly to $y = 10$.

• To solve $3p - 12 = 9p$, subtract $3p$ from both sides to get $-12 = 6p$. Then, divide by 6 to find $p = -2$.

Property

Step 1. Choose one side to be the variable side and then the other will be the constant side.
Step 2. Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality.
Step 3. Collect the constants to the other side, using the Addition or Subtraction Property of Equality.
Step 4. Make the coefficient of the variable 1, using the Multiplication or Division Property of Equality.
Step 5. Check the solution by substituting it into the original equation.

It is a good idea to make the variable side the one in which the variable has the larger coefficient. This usually makes the arithmetic easier.

Examples

• Given $9x + 3 = 4x + 23$, subtract $4x$ from both sides to get $5x + 3 = 23$. Then subtract 3 to get $5x = 20$, so $x = 4$.