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Solving Linear Equations

Students solve one-step and multi-step linear equations by using inverse operations while keeping both sides balanced.

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Beginner 15–20 minutes Solving & Graphing Equations
Solving & Graphing Equations Page 1 of 2
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At a Glance

Solving Linear Equations
Binder SectionSolving & Graphing Equations
DifficultyBeginner
Estimated Time15–20 minutes
PrerequisitesNone
Essential Question

How can we isolate an unknown while keeping an equation balanced?

Build Understanding

An equation states that two expressions are equal. Solving an equation means finding every value of the variable that makes the equation true. First use substitution to decide whether a proposed value is a solution.

Think of an equation as a balanced scale. Whatever operation you perform on one side must also be performed on the other side. Use inverse operations to isolate the variable: addition undoes subtraction, subtraction undoes addition, multiplication undoes division, and division undoes multiplication.

One-step example: x + 7 = 19. Subtract 7 from both sides: x = 12. Check: 12 + 7 = 19.

Two-step example: 3x - 5 = 16. Add 5 to both sides: 3x = 21. Divide both sides by 3: x = 7. Check: 3(7) - 5 = 16. The same reasoning works with positive and negative rational coefficients, including fractions and decimals.

For equations with variables on both sides, move variable terms to one side and constants to the other. Solve 5x + 2 = 3x + 18: subtract 3x to get 2x + 2 = 18; subtract 2 to get 2x = 16; divide by 2, so x = 8.

Distribute and combine like terms before isolating the variable. Some equations have no solution because the variable terms cancel and leave a false statement. Others have infinitely many solutions because they simplify to a true statement.

Equations also model real situations. Define the variable, translate the quantities into an equation, solve, and interpret the answer with units. Literal equations use the same balance rules to isolate one variable in a formula. For example, solve d = rt for t by dividing both sides by r: t = d/r, where r is not zero.

Visual Model
4Thousands 3Hundreds 2Tens 7Ones

4,327 = 4,000 + 300 + 20 + 7

Goals

Learning Targets

  • Use substitution to check a possible solution.
  • Simplify each side first.
  • Use inverse operations and perform the same operation on both sides.
  • Solve equations with rational coefficients and variables on both sides.
  • Recognize one solution, no solution, and infinitely many solutions.
  • Write equations for real situations and rearrange formulas.
  • Check the solution in the original equation.
Click each word

Interactive Vocabulary

Reveal one step at a time

Worked Examples

Example 1

Solve 3x+7=25

1

Subtract 7 from both sides.

Example 2

Solve 4(x−2)+3=19

1

Distribute 4.

Ask and explain

Common Questions

Why must I do the same thing to both sides? To preserve the equality.
Which term should I move first? Choose a step that makes isolating the variable efficient and keeps the work clear.
Can I divide before adding or subtracting? Only when that operation correctly undoes the equation structure.
What if the variables cancel? A true statement means infinitely many solutions; a false statement means no solution.
How do I know my answer is correct? Substitute it into the original equation.
Notice and correct

Common Misconceptions

Changing only one side

Apply the same operation to both sides to preserve equality.

Losing a negative during distribution

Multiply the outside factor by every term, including each sign.

Stopping without checking

Substitute the solution into the original equation.

Learn Through Video

Watch, pause, and explain

Use the chapter markers, key vocabulary, and reflection prompts to stay actively engaged while you watch.

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How to Solve One-Step Equations - Math with Mr. J

1 How to Solve One-Step Equations - Math with Mr. J 2 Solving Two-Step Equations - Math with Mr. J 3 Solving Multi-Step Equations - Math with Mr. J 4 Algebraic Equations Playlist - Math with Mr. J
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Build Your Skills

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Warm-Up Guided Independent Challenge
1
Quick Start

Warm-Up

Warm-up 1

Solve x+9=17.

2
We Do

Guided Practice

Guided 1

Does x = 4 satisfy 3x + 2 = 14? Answer: yes.

Guided 2

Solve x + 9 = 24. Answer: x = 15.

Guided 3

Solve 3x - 8 = 19. Answer: x = 9.

Guided 4

Solve 5(x - 2) = 30. Answer: x = 8.

Guided 5

Solve 7x + 4 = 3x + 28. Answer: x = 6.

Guided 6

Solve 2(x + 5) = 2x + 10. Answer: infinitely many solutions.

Guided 7

Solve 4(x - 1) = 4x + 3. Answer: no solution.

Guided 8

A gym charges $18 plus $7 per class. Write and solve an equation for 6 classes. Answer: C = 18 + 7(6) = $60.

Guided 9

Solve d = rt for t. Answer: t = d/r, r ≠ 0.

3
You Do

Independent Practice

Independent 1

Solve 7x+4=39.

Independent 2

Solve 5x−8=2x+13.

Independent 3

Solve 2(3x−1)=22.

4
Stretch Your Thinking

Challenge & Real-World Practice

🧩 Challenge

Create a multi-step equation whose solution is −3 and includes distribution and variables on both sides. Solve it two ways and verify the result.

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Khan Academy

DeltaMath

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Success Criteria

Before you begin, I can…

  • I can name the inverse operation used at each step.
  • I can preserve equality by changing both sides.
  • I can simplify accurately and verify the solution in the original equation.
1
Question 1

Solve x+8=15.

2
Question 2

Solve 4x=28.

3
Question 3

Solve 3x−5=16.

4
Question 4

Solve 2(x+3)=18.

5
Question 5

What best verifies a solution?

Confidence Check

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Lesson Complete

Great job on Solving Linear Equations!

You reviewed the lesson, practiced the skill, and completed the mastery check.

Next Lesson: Graphing Linear Equations → Back to Binder Topics

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