Introduction

Solving Equations with the Distributive Property is an important Grade 7 math skill because students are moving from simple answers toward explaining how the math works.

In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with solving equations with the distributive property.

What Is Solving Equations with the Distributive Property?

Solving Equations with the Distributive Property means choosing a model, naming what each number means, and explaining the strategy.

The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.

Understanding Solving Equations with the Distributive Property

Before solving, students should slow down and decide what each number, shape, unit, or label represents.

  • Read the question carefully and identify what is being asked.
  • Choose a model, equation, table, or diagram that matches the situation.
  • Solve one step at a time and keep units or labels attached.
  • Use the answer explanation to check that the result makes sense.

Visual Models

Visual Model 1

Question: Solve for \(x\): \(3(2x+1)=21\)

Visual Model 1

  • A. \(x=2\)
  • B. \(x=3\)
  • C. \(x=4\)
  • D. \(x=5\)

Why it works: Distribute: \(6x+3=21\). Subtract \(3\): \(6x=18\). Divide by \(6\): \(x=3\).

Answer: \(x=3\)

Visual Model 2

Question: Solve for \(x\): \(4(2x-3)=20\)

Visual Model 2

  • A. \(x=1\)
  • B. \(x=2\)
  • C. \(x=3\)
  • D. \(x=4\)

Why it works: Distribute: \(8x-12=20\). Add \(12\): \(8x=32\). Divide by \(8\): \(x=4\).

Answer: \(x=4\)

Worked Examples

Example 1

Question: Which equation is equivalent to \(2(x+8)=30\)?

Example 1

  • A. \(2x+8=30\)
  • B. \(2x+16=30\)
  • C. \(x+16=30\)
  • D. \(2x=38\)
  1. The distributive property gives \(2 \cdot x + 2 \cdot 8 = 30\), so \(2x+16=30\).

Answer: \(2x+16=30\)

Example 2

Question: A rectangle has width 3 and length \(x+2\). If the area is \(30\) square units, what is the length?

Example 2

  • A. \(x+2=8\)
  • B. \(x+2=10\)
  • C. \(x+2=12\)
  • D. \(x+2=15\)
  1. Area: \(3(x+2)=30\).
  2. Divide by \(3\): \(x+2=10\).
  3. (So length is \(10\) units; if needed, \(x=8\).)

Answer: \(x+2=10\)

Example 3

Question: Solve for \(x\): \(3(x+4)=27\)

  • A. \(x=5\)
  • B. \(x=7\)
  • C. \(x=9\)
  • D. \(x=23\)
  1. Distribute: \(3x+12=27\).
  2. Subtract \(12\): \(3x=15\).
  3. Divide by \(3\): \(x=5\).
  4. (Distractor D: forgetting to distribute gives \(x+4=27 \Rightarrow x=23\).)

Answer: \(x=5\)

Real-World Word Problems

Problem 1

Question: A shop charges $\(5\) per pizza and then adds a delivery fee. If 4 pizzas plus delivery cost $\(28\) total, what is the delivery fee?

  • A. $\(3\)
  • B. $\(5\)
  • C. $\(8\)
  • D. $\(12\)

Why it works: Let \(d\) be the delivery fee. Then \(4(5) + d = 28\), so \(20 + d = 28\), giving \(d = 8\).

Answer: Delivery fee is $\(8\)

Problem 2

Question: A teacher buys 5 notebooks at $\(x\) each and spends a total of $\(45\) (including a $\(5\) coupon discount). What is the price per notebook?

  • A. $\(8\)
  • B. $\(9\)
  • C. $\(10\)
  • D. $\(12\)

Why it works: Equation: \(5x - 5 = 45\). Add \(5\): \(5x = 50\). Divide by \(5\): \(x = 10\).

Answer: Price per notebook is $\(10\)

Common Mistakes

  • Rushing before identifying what the numbers represent.
  • Choosing an operation that does not match the situation.
  • Dropping labels, units, or context from the answer.
  • Skipping the estimate or reasonableness check.

Strategy Tips

  • Underline the question being asked.
  • Use a model before jumping to computation.
  • Write an equation that matches the story or picture.
  • Explain the final answer in a sentence.

Practice Questions

Question 1

Solve for \(x\): \(5(x-2)=35\)

  • A. \(x=5\)
  • B. \(x=7\)
  • C. \(x=9\)
  • D. \(x=12\)

Question 2

Solve for \(x\): \(2(x+6)=20\)

  • A. \(x=2\)
  • B. \(x=4\)
  • C. \(x=6\)
  • D. \(x=8\)

Question 3

Solve for \(x\): \(4(x-3)=12\)

  • A. \(x=0\)
  • B. \(x=3\)
  • C. \(x=6\)
  • D. \(x=9\)

Question 4

Solve for \(x\): \(6(x+2)=30\)

  • A. \(x=1\)
  • B. \(x=3\)
  • C. \(x=5\)
  • D. \(x=7\)

Question 5

Solve for \(x\): \(2(x-5)=14\)

  • A. \(x=7\)
  • B. \(x=12\)
  • C. \(x=9\)
  • D. \(x=5\)

Question 6

Solve for \(x\): \(7(x+1)=49\)

  • A. \(x=5\)
  • B. \(x=6\)
  • C. \(x=7\)
  • D. \(x=8\)
Full Answer Explanations Click to show all answers and explanations

Question 1

Answer: \(x=9\)

Distribute: \(5x-10=35\). Add \(10\): \(5x=45\). Divide by \(5\): \(x=9\). (Distractor A: solving \(x-2=5\) instead; Distractor D: arithmetic error when isolating \(x\).)

Question 2

Answer: \(x=4\)

Distribute: \(2x+12=20\). Subtract \(12\): \(2x=8\). Divide by \(2\): \(x=4\). (Distractor A: dividing only the right side; Distractor C: forgetting to divide by 2; Distractor D: arithmetic error in final step.)

Question 3

Answer: \(x=6\)

Distribute: \(4x-12=12\). Add \(12\): \(4x=24\). Divide by \(4\): \(x=6\).

Question 4

Answer: \(x=3\)

Distribute: \(6x+12=30\). Subtract \(12\): \(6x=18\). Divide by \(6\): \(x=3\).

Question 5

Answer: \(x=12\)

Distribute: \(2x-10=14\). Add \(10\): \(2x=24\). Divide by \(2\): \(x=12\).

Question 6

Answer: \(x=6\)

Distribute: \(7x+7=49\). Subtract \(7\): \(7x=42\). Divide by \(7\): \(x=6\).

Timed practice quizzes

Related Quizzes

After studying Solving Equations with the Distributive Property, open these two timed quizzes in a new tab for focused extra practice.

Connection to Standards

This lesson supports Grade 7 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.

Summary

Solving Equations with the Distributive Property becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.

GOLDEN RULE

Understand the model before choosing the operation.