3.01 Multi-step equations

Lesson

Practice

Introduction

We have already learned how to solve two-step equations that require a combination of multiplication, division, subtraction, and/or adddition processes to find the value of the variable. We are now ready to solve multi-step equations which include equations whose solutions require expanding expressions using the distributive property and collecting like terms.

Ideas

Solving multi-step equations

Solving multi-step equations

To solve multi-step equations involving parentheses, we will use the distributive property and properties of equality to remove parentheses and get the variable by itself on one side of the equals sign.

We can follow these steps to solve one-variable equations involving parentheses.

Use the distributive property to remove parentheses.
Combine like terms to simplify each side of the equals sign.
Add or subtract to get the variable term on one side of the equals sign.
Multiply or divide to isolate the variable and solve the equation.

Examples

Example 1

Solve the following equation: - 5 \left(x + 4\right) + 9 = - 41

Worked Solution

Create a strategy

Use the distributive property and properties of equality to remove the parentheses and isolate the variable.

Apply the idea

\displaystyle - 5x-20 + 9	\displaystyle =	\displaystyle -41	Use distributive property to expand -5(x+4)
\displaystyle -5x-20+9+20-9	\displaystyle =	\displaystyle -41+20-9	Add 20 and subtract 9 to both sides
\displaystyle -5x	\displaystyle =	\displaystyle -30	Simplify
\displaystyle \frac{-5x}{-5}	\displaystyle =	\displaystyle \frac{-30}{-5}	Divide both sides by -5
\displaystyle x	\displaystyle =	\displaystyle 6	Evaluate

Example 2

Solve the following equation: 5 \left( 2 y + 2\right) + 3 \left( 4 y - 5\right) + 5 y = 45

Worked Solution

Create a strategy

Use the distributive property and properties of equality to remove the parentheses and isolate the variable. Combine like terms.

Apply the idea

\displaystyle 10y+10+12y-15+5y	\displaystyle =	\displaystyle 45	Use distributive property
\displaystyle 27y-5	\displaystyle =	\displaystyle 45	Combine like terms
\displaystyle 27y-5+5	\displaystyle =	\displaystyle 45+5	Add 5 to both sides
\displaystyle 27y	\displaystyle =	\displaystyle 50	Simplify
\displaystyle \frac{27y}{27}	\displaystyle =	\displaystyle \dfrac{50}{27}	Divide both sides by 27
\displaystyle x	\displaystyle =	\displaystyle \dfrac{50}{27}	Evaluate

Idea summary

When solving multi-step equations involving parentheses:

Use the distributive property to clear parentheses.
Combine like terms to simplify each side.
Add or subtract to get the variable term on one side of the equals sign.
Multiply or divide to isolate the variable and solve the equation.

Outcomes

8.EE.C.7

Solve linear equations in one variable.

8.EE.C.7.B

Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.