Lesson 3: Solve Equations with Variables and Constants on Both Sides

New Concept

This lesson provides a universal strategy for solving linear equations. You'll learn to gather all variable terms on one side and constants on the other, transforming any equation into the simpler form $ax = b$ to find the solution.

What’s next

Next, you'll work through interactive examples and practice cards, moving from simpler to more complex equations and mastering the general strategy for solving.

Property

Our strategy will involve choosing one side of the equation to be the variable side, and the other side of the equation to be the constant side. Then, we will use the Subtraction and Addition Properties of Equality, step by step, to get all the variable terms together on one side of the equation and the constant terms together on the other side. By doing this, we will transform the equation that started with variables and constants on both sides into the form $ax = b$.

Examples

• To solve $3x + 5 = 20$, subtract 5 from both sides to get $3x = 15$. Then, divide by 3 to find $x = 5$.

• For the equation $5a - 9 = 21$, add 9 to both sides, which gives $5a = 30$. Dividing both sides by 5 results in $a = 6$.

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 $x = 4$.

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 to one side of the equation. Use the Addition or Subtraction Property of Equality.
Step 3. Collect all the constant terms to 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

• To solve $5(y - 4) = 15$, first distribute to get $5y - 20 = 15$. Add 20 to both sides for $5y = 35$, so $y = 7$.

• For $2(a - 3) + 7 = 9$, first distribute and combine like terms: $2a - 6 + 7 = 9$, which simplifies to $2a + 1 = 9$. Then $2a = 8$, so $a = 4$.