Section 1.3: Solving Equations with Variables on Both Sides

Property

Step 1. Simplify each side of the equation as much as possible.

• Use the Distributive Property to remove any parentheses.
• Combine like terms.

Step 2. Collect all the variable terms on one side of the equation.

• Use the Addition or Subtraction Property of Equality.

Step 3. Collect all the constant terms on the other side of the equation.

• Use the Addition or Subtraction Property of Equality.

Step 4. Make the coefficient of the variable term to equal to 1.

• Use the Multiplication or Division Property of Equality.
• State the solution to the equation.

Step 5. Check the solution.

• Substitute the solution into the original equation to make sure the result is a true statement.

Examples

Property

For equations with variables on both sides of the equation, begin as we did above—choose 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.

Examples

• Solve $10x = 9x - 7$. Subtract $9x$ from both sides to get all the variables on the left, which gives the solution $x = -7$.

• Solve $3y - 8 = 6y$. Subtract $3y$ from both sides to gather the variables on the right, which gives $-8 = 3y$. Divide by 3 to get $y = -\frac{8}{3}$.

Property

An equation has infinitely many solutions when algebraic manipulation results in a true statement where both sides are identical, such as $a = a$ where $a$ is any real number.

Examples