Solving Two-Step Equations

Chapter 3: Multi-Step Equations and Inequalities

Lesson 3.1: Solving Two-Step Equations

Use two inverse operations to solve equations with two steps.

Learning Objectives

By the end of this lesson, you should be able to:

Solve two-step equations using inverse operations.
Write, solve, and check two-step equations that model real-life situations.

Jesus Searches for the Lost

Some equations require two careful steps. We first undo the addition or subtraction, and then undo the multiplication or division. Following the correct order helps us find the value that was hidden.

Jesus often used parables to teach about God’s love. In the Parable of the Lost Sheep, a shepherd leaves the ninety-nine sheep to search for one sheep that is lost. When he finds it, he rejoices and invites others to celebrate. Jesus taught that God cares deeply for every person and lovingly seeks those who are far from Him.

This parable points to Jesus Christ, who came to seek and save the lost. Our sin separates us from God, but Jesus willingly gave His life on the cross, rose again, and offers forgiveness and reconciliation to everyone who turns from sin and trusts in Him. The love of Jesus is personal, patient, and joyful.

As you solve each equation one step at a time, remember that Jesus does not abandon those who are lost. He calls us to receive His love, believe the gospel, and follow Him.

Memory verse: “Rejoice with me, for I have found my sheep that was lost.” — Luke 15:6

1. What Is a Two-Step Equation?

A two-step equation is an equation that requires two inverse operations to isolate the variable. For example, in $3x+7=-5$ , the variable is multiplied by 3 and then 7 is added.

General Strategy

Undo the addition or subtraction first.
Undo the multiplication or division second.
Check the solution in the original equation.

$ax+b=c \quad\Longrightarrow\quad ax=c-b \quad\Longrightarrow\quad x=\frac{c-b}{a}$

The order matters because the operations were performed on the variable in the opposite order. First remove the constant term, then remove the coefficient.

2. Solving by Subtraction and Division

Worked Example 1

Solve $3x+7=-5$ .

Step 1: Subtract 7 from both sides.

$3x+7-7=-5-7$

$3x=-12$

Step 2: Divide both sides by 3.

$x=-4$

Check: $3(-4)+7=-12+7=-5$ ✓

Important Reminder

When subtracting a positive number from both sides, subtract it from both sides. When the constant is negative, adding its opposite may be easier.

$-2q-63=47 \quad\Longrightarrow\quad -2q=110 \quad\Longrightarrow\quad q=-55$

3. Solving by Addition and Multiplication

Worked Example 2

Solve

$\frac{x}{2}-3=1.$

Step 1: Add 3 to both sides.

$\frac{x}{2}-3+3=1+3$

$\frac{x}{2}=4$

Step 2: Multiply both sides by 2.

$x=8$

Check:

$\frac{8}{2}-3=4-3=1$

✓

Fraction Form

A fraction such as $\dfrac{x}{2}$ means that $x$ is divided by 2. To undo the division, multiply both sides by 2.

$\frac{x}{a}+b=c \quad\Longrightarrow\quad \frac{x}{a}=c-b \quad\Longrightarrow\quad x=a(c-b)$

4. Solving Equations with Negative Coefficients

Worked Example 3

Solve $7-4y=19$ .

Step 1: Subtract 7 from both sides.

$7-4y-7=19-7$

$-4y=12$

Step 2: Divide both sides by $-4$ .

$y=-3$

Check: $7-4(-3)=7+12=19$ ✓

5. Writing and Solving a Two-Step Equation

Worked Example 4: A Drum Set

A drum set costs 495 dollars. You make a down payment of 150 dollars and pay the remaining cost in three equal monthly payments. How much is each monthly payment?

Let $p$ represent one monthly payment.

$150+3p=495$

Subtract 150 from both sides:

$3p=345$

Divide both sides by 3:

$p=115$

Answer: Each monthly payment is 115 dollars. Check: $150+3(115)=495$ ✓

Four-Step Problem-Solving Routine

Define: Let a variable represent the unknown.
Translate: Write an equation from the situation.
Solve: Use two inverse operations.
Check: Substitute the answer and decide whether it makes sense.

Common Mistakes to Avoid

Do not perform only one of the two steps.
Undo addition or subtraction before undoing multiplication or division.
Apply every operation to both sides of the equation.
Be especially careful when dividing by a negative coefficient.
Always check the answer in the original equation.

Practice Questions

Use two inverse operations to solve each equation. Show your steps and check your solutions.

1. Skill Practice

$12k+7=31$
$13n+42=81$
$56=17p-29$
$\frac{w}{4}-21=-3$
$\frac{h}{9}-19=-10$
$\frac{d}{12}+25=29$
$12=\frac{a}{36}+17$
$18-r=42$
$80=23-3v$
$-2q-63=47$
$-\frac{x}{2}+4=12$
$-5=-19-\frac{x}{7}$

Write the verbal sentence as an equation. Then solve the equation.

Five minus the product of 2 and a number is 7.
Thirty-two minus the product of 9 and a number is 140.
Thirteen plus the product of 6 and a number is 67.
Negative 8 minus the product of 3 and a number is 19.

2. Applications

Driving. Your family begins a long-distance car trip with 16 gallons of gasoline. The car uses 3 gallons per hour. You will stop to refuel when exactly 1 gallon remains.
1. List the information you are given and the information you need to find.
2. Write a verbal model and an equation.
3. After how many hours will you need to refuel? Justify your solution with a table.
Rafting. A group of 9 friends takes a white-water rafting trip. The total price before discounts is 810 dollars. The total price after student discounts is 729 dollars. How much is the discount per person?
Trains. A train consisting of 50 cars and one locomotive weighs 4725 tons. The locomotive weighs 125 tons. All the cars have the same weight. Find the weight of one car.

3. Additional Applications

Taxi Fare. A taxi charges a fixed fee of 4.50 euros plus 1.80 euros per kilometer. If the total fare is 22.50 euros, how many kilometers did the passenger travel?
Savings. A student already has 35 euros and saves 12 euros each week. How many weeks will it take to save 119 euros?
Temperature. The temperature is $-4^\circ\text{C}$ at sunrise and increases by $3^\circ\text{C}$ each hour. After how many hours will the temperature reach $20^\circ\text{C}$ ?
Perimeter. A rectangle has a width of 6 meters and a perimeter of 34 meters. Find its length.
Fundraising. A class pays a fixed registration fee of 15 euros and collects 8 euros from each student. If the total collected is 95 euros, how many students contributed?
Phone Plan. A phone plan charges a monthly fee of 25 euros plus 5 euros for each extra gigabyte of data. If the bill is 60 euros, how many extra gigabytes were used?