Lesson 3: Solving Equations with Variables on Both Sides

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 $x = 4$.

Property

An equation that is true for any value of the variable is called an identity. The solution of an identity is all real numbers.

Examples

• The equation $3(x+2) = 3x+6$ is an identity. If we distribute on the left, we get $3x+6 = 3x+6$. This simplifies to $6=6$, which is always true. The solution is all real numbers.

• Consider $7 + 4(m-1) = 4m + 3$. Simplifying the left side gives $7+4m-4 = 4m+3$, which becomes $4m+3 = 4m+3$. This is an identity, and the solution is all real numbers.