Lesson 2: Solve 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

When a negative sign is in front of a parenthesis, it must be distributed to every term inside, changing the sign of each term.

Examples

Property

An alternate method to solve equations with fractions is to eliminate the fractions.
We will apply the Multiplication Property of Equality and multiply both sides of an equation by the least common denominator of all the fractions in the equation.
The result of this operation will be a new equation, equivalent to the first, but without fractions.
This process is called “clearing” the equation of fractions.

Strategy to solve equations with fraction coefficients.
Step 1. Find the least common denominator of all the fractions in the equation.
Step 2. Multiply both sides of the equation by that LCD. This clears the fractions.
Step 3. Solve using the General Strategy for Solving Linear Equations.

Examples

• Solve $\frac{1}{5}y - \frac{1}{2} = \frac{3}{10}$. The LCD is 10. Multiplying by 10 gives $10(\frac{1}{5}y) - 10(\frac{1}{2}) = 10(\frac{3}{10})$, which simplifies to $2y - 5 = 3$. Solving gives $y = 4$.