Common Core (CCSS)8th GradeMath

8.EE.C.7— Solving Linear Equations in One Variable

Expressions & Equations · 8.EE.C: Analyze and solve linear equations and pairs of simultaneous linear equations.

$Page 1 of the free 8.EE.C.7 worksheet PDF$

Free printable worksheet

Download the 8.EE.C.7 worksheet

The lesson, the visual models, and 16 practice questions with space to work.

PDF · 10 pages · answer key · no sign-up

What this standard means: Eighth graders learn to solve linear equations in one variable and to explain why each step is allowed.

Students start with equations where the variable sits on only one side, then move to harder ones where it appears on both sides and has to be collected first. Along the way they expand brackets, combine like terms, and keep both sides balanced.

They also meet the three things that can happen at the end: exactly one solution, no solution at all, or infinitely many solutions because the two sides are really the same expression in disguise.

Recognizing those three outcomes, and being able to say which one an equation has and why, is what separates a student who understands equations from one who is only following memorized steps.

What the standard says, word for word · 8.EE.C.7

Solve linear equations in one variable

Read it at the source: Common Core State Standards

What’s inside

What you get, and who it’s for

Everything below is measured from the actual lesson and the actual file. Nothing here is an estimate.

Time to run: About 45 minutes. 15 min lesson, then 30 min of practice.
In the download: 10 printable pages: the full lesson, the worked examples, the visual models, and 16 practice questions with space to work. An educator answer key is included at the end, so the student pages come first and it is still ready to hand out. The lesson’s worked examples do show their solutions, because that is how they teach.
Question set: The download carries all 16 questions. The page shows 16 of them as a sample.
Question format: Every question is numeric answer. Students show their work on the page.
Assumed before this: Students who are shaky on these will struggle with 8.EE.C.7. Worth a quick check first.
7.EE.B.4(Grade 7) Students should already be able to use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities involved.
Dates: Published August 5, 2026. Last updated August 14, 2026.

The sample intervention

See a real lesson for 8.EE.C.7

A real intervention, not a generic worksheet.

8.EE.C.7

Solve for the Unknown: Equation Practice Mix

Solve linear equations in one variable using balancing steps and checking for special cases.

Read the lesson first, then work through the practice set. You will solve equations with the variable on one side and equations with the variable on BOTH sides, and you will meet all three possible outcomes: exactly one solution, no solution, or infinitely many solutions. Show every balancing step, and check each answer by putting it back into the original equation.

A balance scale with its beam level, both pans at the same height.

A linear equation in one variable is a number sentence with one letter, such as x, that stands for an unknown number. To solve it, keep the equation balanced by doing the same operation to both sides. Each move should undo the operation attached to the variable until the letter is alone.

Think of a balance scale: if one side changes, the other side must change in the same way. For example, if 3x + 4 = 19, subtract 4 from both sides, then divide both sides by 3. The equation stays balanced at every step, and the final value tells you the unknown number.

Balance the equation

3x + 4 = 19, the same equation worked step by step in Example 1.

Worked exampleExample 1: 3x + 4 = 19 → subtract 4 from both sides: 3x = 15 → divide both sides by 3: x = 5.

Some equations have the variable on both sides. In those cases, collect the variable terms on one side first. For example, 5x + 2 = 2x + 11 becomes 3x + 2 = 11 after subtracting 2x from both sides. Then continue solving by undoing the remaining operations.

Variables on BOTH sides

Here x sits on both pans. Take 2x off both sides first, then finish as in Example 2.

Worked exampleExample 2: 5x + 2 = 2x + 11 → subtract 2x from both sides: 3x + 2 = 11 → subtract 2: 3x = 9 → divide by 3: x = 3.

Worked exampleExample 3: 2(x + 4) = 18 → divide both sides by 2: x + 4 = 9 → subtract 4: x = 5.

What "distribute" really means

2(x + 4)

Two groups of (x + 4): 2 x-tiles and 8 units, so 2x + 8. BOTH terms get multiplied, and the same is true when the second term is negative.

Sometimes an equation has no solution or infinitely many solutions. If simplifying leaves a false statement, like 2 = 7, there is no solution. If simplifying leaves a true statement, like 4 = 4, there are infinitely many solutions. These cases show that not every equation has exactly one answer.

Worked exampleExample 4: 4x + 6 = 4x + 10 → subtract 4x from both sides: 6 = 10, so there is no solution.

Do 4x + 6 and 4x + 10 ever meet?

	4x + 6	4x + 10
x = 0	6	10
x = 1	10	14
x = 2	14	18
x = 3	18	22

The two sides stay exactly 4 apart, so no value of x makes them equal.

Worked exampleExample 5: 3(2x - 1) = 6x - 3 → distribute: 6x - 3 = 6x - 3 → subtract 6x from both sides: -3 = -3, so there are infinitely many solutions.

Real-life situations like gym memberships and ticket pricing often form linear equations. A gym may charge a start-up fee plus a monthly rate, or tickets may cost a fixed amount per person. Solving the equation tells you the unknown number of months, tickets, or total cost.

Practice questions

1
Solve for x: 2x + 5 = 17.
2
A student solved 4x - 3 = 13 like this: 4x = 10, so x = 2.5. What is the correct value of x?
3
A ticket package costs $9 plus $4 per ticket. The total is $29. How many tickets were bought?
4
Solve for x: 6x + 4 = 2x + 20.
5
Which equation has the larger solution: 4x + 2 = 18 or 2x + 8 = 18? Enter the larger x value.
6
A student says 4x + 6 = 4x + 10 has x = 1 because the x terms match. What is the correct answer category? Enter 0 for no solution or -1 for infinitely many solutions.
7
A membership starts with $8 and adds $6 each month. The equation is 6x + 8 = 32. What does x equal?
8
Solve for x: 3(2x - 4) = 18.
9
A gym charges a $15 start-up fee plus $10 per month. The total cost is $55. How many months did the person pay for?
10
A student solved 2(x - 3) = 14 like this: 2x - 3 = 14, so x = 8. What is the correct value of x?
11
Compare the solutions. Which x is smaller: 3x + 9 = 24 or 5x + 1 = 31? Enter the smaller solution.
12
A student says 3(x + 2) = 18 means x + 2 = 18 and x = 16. What is the correct value of x?
13
Solve for x: 7x - 9 = 26.
14
Which equation has the larger x value: 2x + 10 = 30 or 6x - 6 = 24? Enter the larger x value.
15
A gym plan charges $18 per month with a $12 sign-up fee. If the total is $66, how many months were paid for?
16
A gym charges a one-time signup fee plus a monthly membership fee. After 3 months, the total cost is $47. If the signup fee is $8, what is the monthly fee, in dollars?

Part of the question set is shown. Generate the whole intervention, freshly themed and with an answer key, for your own class in the product.

Watch for this

Common mistakes

Skipping a step when undoing operations

The most common mistake is changing only one part of the equation or undoing the wrong operation, instead of keeping both sides balanced. Students may also divide too soon or miss a factor or negative sign, which leads to answers like x = 8 for 2(x - 3) = 14 or x = 2.5 for 4x - 3 = 13. The fix is to show each balancing step, one operation at a time, and then check the result in the original equation.

Vocabulary

Key terms

linear equation in one variable: A number sentence with one letter, such as x, that stands for an unknown number.
balanced equation: An equation that stays equal when you do the same operation to both sides.
variable terms: The parts of an equation that include the variable.
solution: The value that makes an equation true when you put it back in the original equation.

In the classroom

How to use this intervention

Use the 15-minute lesson for a short whole-class warm-up and model one or two equations, including a case with variables on both sides. Use the 30-minute practice set mainly for independent work, then pull a small group to check balancing steps, answer checks, and special cases like no solution or infinitely many solutions.

Want this themed to your class instead?

Generate a fresh, standards-aligned intervention like this one, themed to whatever your students are into, in about 2 minutes.

Generate for my class

Questions educators ask

Educator FAQ

What grade is this worksheet for?

It is for Grade 8.

Which CCSS standard does it align to?

It aligns to CCSS 8.EE.C.7, solving linear equations in one variable.

Is an answer key included?

Yes, the free worksheet PDF includes all questions with a complete answer key.