Solving Linear Equations Worksheet: Free PDF for Every Level
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Last Updated: July 2026
In This Guide
- Introduction: Two Levels, One Guide
- Quick Answer
- Download the Free PDF
- TL;DR + Quick Facts
- What Is a Linear Equation?
- Step-by-Step: How to Solve Any Linear Equation
- Worked Examples (Beginner to Advanced)
- Common Mistakes and How to Fix Them
- Unique Insight: What Most Guides Get Wrong
- On-Page Practice Worksheet
- Quick Quiz
- FAQ
- Key Takeaways
- Related Articles
- About the Author
- Sources & References
Solving linear equations is the single most important algebra skill a student can master. Once you understand how to isolate a variable, every other algebra topic — from systems of equations to quadratics — becomes far easier to learn.
Whether you are a grade 6 student seeing equations for the first time or a grade 10 student preparing for a standardised test, this guide is built for both of you. Each section gives a beginner explanation first, then layers in the advanced version. You get a full lesson, three worked examples, and a free printable worksheet with an answer key.
In my experience tutoring hundreds of students, the gap between “I don’t get algebra” and “I can solve any linear equation” is almost always just one missing concept — and I will show you exactly what that is.
What you will learn in this guide:
- What a linear equation is (and what it is not)
- A four-step method that works for every linear equation
- Three fully worked examples from one-step to multi-step
- The most common mistake students make (and the fix)
- 10 practice problems with a full answer key
A linear equation is an equation where the variable has no exponent greater than 1. To solve it, use inverse operations to isolate the variable: undo addition or subtraction first, then multiplication or division. For example, 2x + 5 = 13 solves to x = 4 in two steps. This worksheet covers one-step through multi-step equations for grades 6-10.
Free Printable PDF: Get the full worksheet (10 problems + answer key) in a clean print-ready format.
⚡ TL;DR – Quick Summary
- A linear equation has a variable with exponent 1 and one solution.
- The goal is always to isolate the variable using inverse operations.
- Four steps: simplify, move variables left, move constants right, divide.
- Always check your answer by substituting it back into the original equation.
- This worksheet has 10 problems, from x + 7 = 15 up to (2x + 6)/4 = 5.
- Download the free PDF or work through the problems on this page.
| Feature | Detail |
|---|---|
| Skill name | Solving linear equations |
| Grade range | Grades 6-10 (middle and high school) |
| Equation types covered | One-step, two-step, multi-step, brackets, fractions |
| Number of problems | 10 (ordered easy to hard) |
| PDF download | Free, includes answer key |
| Key operation | Inverse operations to isolate the variable |
| Check method | Substitute answer back into original equation |
What Is a Linear Equation?
A linear equation is an algebraic equation where the variable (usually x or y) appears with an exponent of exactly 1. The word “linear” comes from the fact that its graph is always a straight line.
Beginner definition: A linear equation is a number sentence with an unknown, like x + 3 = 7. Your job is to find what number x must be to make the sentence true.
Advanced definition: A linear equation in one variable has the standard form ax + b = c, where a, b, and c are real numbers and a is not equal to 0. It has exactly one solution (unless it is an identity or a contradiction).
| Equation | Linear? | Why |
|---|---|---|
| x + 5 = 12 | Yes | Variable exponent is 1 |
| 3x – 7 = 8 | Yes | Variable exponent is 1 |
| x^2 + 3 = 19 | No | Variable is squared (exponent 2) |
| sqrt(x) = 4 | No | Variable is under a square root |
| 2(x + 1) = 10 | Yes | Expands to 2x + 2 = 10 (exponent 1) |
I have seen students panic when they see brackets or fractions in an equation, assuming it is “a different type.” In my experience, it is not. Every linear equation — no matter how it looks — reduces to the same four-step method once you simplify. The brackets and fractions are just cosmetic. That realisation alone cuts student anxiety in half.
How Do You Solve a Linear Equation Step by Step?
You solve a linear equation by applying the same operation to both sides until the variable stands alone. Here is the four-step method that works for every linear equation.
The Four-Step Method
- Simplify both sides. Expand any brackets and combine like terms on each side separately.
- Move all variable terms to the left side. Add or subtract terms containing x so they are all on one side.
- Move all constant terms to the right side. Add or subtract plain numbers so they are all on the other side.
- Isolate the variable. Divide (or multiply) both sides by the coefficient of x.
Solve: 2x + 5 = 13
Step 1 — Simplify: both sides already simplified.
2x + 5 = 13
Step 2 — Move constants right (subtract 5 from both sides):
2x + 5 - 5 = 13 - 5
2x = 8
Step 3 — Isolate x (divide both sides by 2):
2x / 2 = 8 / 2
x = 4
Step 4 — CHECK: substitute x = 4 back in:
2(4) + 5 = 8 + 5 = 13 ✓
Pro Tip: The Balance Scale Model
Think of an equation as a balance scale. Whatever you do to one side, you must do to the other side to keep it balanced. This mental model makes every step feel logical rather than arbitrary.
Worked Examples: Beginner to Advanced
Here are three fully worked examples, one at each difficulty level. Study each one before attempting the worksheet problems.
Example 1 (Level 1 — One-Step): x + 7 = 15
Goal: Undo the +7 by subtracting 7 from both sides.
x + 7 – 7 = 15 – 7
x = 8
Check: 8 + 7 = 15. Correct.
Example 2 (Level 2 — Two-Step): 4y – 7 = 13
Step 1: Add 7 to both sides to remove the constant.
4y – 7 + 7 = 13 + 7 → 4y = 20
Step 2: Divide both sides by 4.
4y / 4 = 20 / 4 → y = 5
Check: 4(5) – 7 = 20 – 7 = 13. Correct.
Example 3 (Level 3 — Brackets): 2(3x + 1) = 4x + 10
Step 1 — Expand brackets: 6x + 2 = 4x + 10
Step 2 — Move variable terms left: 6x – 4x + 2 = 10 → 2x + 2 = 10
Step 3 — Move constants right: 2x = 10 – 2 → 2x = 8
Step 4 — Divide: x = 4
Check: 2(3(4) + 1) = 2(13) = 26. Right side: 4(4) + 10 = 26. Correct.
[IMAGE: Three-column worked examples comparison chart showing one-step, two-step, and multi-step equations side by side | ALT: comparison of one-step two-step and multi-step linear equation worked examples for students]
Common Mistakes Students Make When Solving Linear Equations
These four errors account for the majority of wrong answers I see on student worksheets. Recognising them is half the battle.
Mistake 1: Forgetting to apply the operation to BOTH sides
Wrong: 2x + 3 = 11 → 2x = 11 (only subtracted 3 from the left)
Right: 2x + 3 – 3 = 11 – 3 → 2x = 8
The equation is a balance. Subtract from one side, subtract from both.
Mistake 2: Dividing only the first term when expanding brackets
Wrong: 5(x – 2) = 15 → 5x – 2 = 15 (forgot to multiply -2 by 5)
Right: 5(x – 2) = 15 → 5x – 10 = 15
The distributive property applies to every term inside the bracket.
Mistake 3: Sign errors when moving terms across the equals sign
Wrong: 3x + 5 = 2x + 9 → 3x + 2x = 9 + 5 (moved 2x but kept its sign)
Right: 3x – 2x = 9 – 5 → x = 4
When you move a term across the equals sign, its sign flips.
Mistake 4: Skipping the check step
Students who skip substituting their answer back in miss their own arithmetic errors. The check step takes 20 seconds and catches most mistakes before they cost marks.
Mistake 3 — the sign error — is by far the most common one I see, even among students who understand the method. My fix: instead of “moving a term across,” always think “I am adding the opposite to both sides.” That framing eliminates the sign flip confusion entirely because you are never moving anything; you are just adding.
💡 Unique Insight: The Real Reason Students Struggle With Multi-Step Equations
Most guides teach the steps correctly but miss the underlying reason students stall at multi-step equations: they try to do two operations in one step to save time, and that is where errors compound. In my experience, students who write out every single step — even the “obvious” ones — make fewer errors, not more. The step-skipping habit forms because teachers praise speed. But on a test, one skipped step that hides a sign error costs more marks than the time saved.
The non-obvious fix: teach students to narrate each step aloud or in writing (“I am subtracting 5 from both sides because 5 is being added to x”). This metacognitive habit forces full attention on each operation and reduces sign errors by a significant margin. No worksheet on the first page of Google mentions this.
On-Page Practice Worksheet: 10 Problems (Easy to Hard)
Work through these 10 problems using the four-step method. Show your working for each one. When you are done, click “Show Answer Key” to check your answers.
How to Use This Worksheet
Print the PDF (button below) or write your answers on paper. Solve each equation for the unknown variable. Show every step. After finishing, use the answer key to self-check. For any problem you got wrong, go back to the worked example at the same difficulty level and identify which step you missed.
- x + 7 = 15
- y – 4 = 11
- 3x = 24
- x/5 = 6
- 2x + 3 = 11
- 4y – 7 = 13
- 3x + 5 = 2x + 9
- 5(x – 2) = 15
- 2(3x + 1) = 4x + 10
- (2x + 6)/4 = 5
Show Answer Key
- x = 8
- y = 15
- x = 8
- x = 30
- x = 4
- y = 5
- x = 4
- x = 5
- x = 4
- x = 7
Show worked solution for Problem 10 (hardest)
Solve: (2x + 6)/4 = 5
Step 1 — Multiply both sides by 4: 2x + 6 = 20
Step 2 — Subtract 6 from both sides: 2x = 14
Step 3 — Divide both sides by 2: x = 7
Check: (2(7) + 6)/4 = (14 + 6)/4 = 20/4 = 5. Correct.
Want a print-ready version? Download the full PDF with all 10 problems and a separate answer key section — perfect for classroom or home use.
Quick Quiz: Test Your Understanding
3-Question Check
Q1. What is the solution to x + 9 = 16?
Q2. What is the first step to solve 3x – 4 = 11?
Q3. Solve: 5(x – 2) = 15. What is x?
Frequently Asked Questions About Solving Linear Equations
What is a linear equation?
A linear equation is an algebraic equation where the variable appears with an exponent of 1. Its graph is a straight line. Examples include x + 3 = 7 and 2x – 5 = 9. It always has exactly one solution (unless it is an identity or contradiction).
What grade level is this worksheet designed for?
This worksheet covers grades 6-10. Problems 1-4 suit grade 6-7 students learning one-step equations. Problems 5-7 target grade 7-8 students working on two-step equations. Problems 8-10 challenge grade 9-10 students with brackets and fractions.
How do I check my answer after solving a linear equation?
Substitute your answer back into the original equation and simplify both sides. If both sides equal the same number, your answer is correct. For 2x + 3 = 11 with x = 4: 2(4) + 3 = 8 + 3 = 11. Both sides match, so x = 4 is correct.
What is the difference between a one-step and a two-step equation?
A one-step equation needs exactly one inverse operation to isolate the variable (e.g., x + 7 = 15 needs one subtraction). A two-step equation needs two operations in sequence (e.g., 2x + 3 = 11 needs subtraction then division). The method is the same; two-step just adds one more operation.
What does “isolate the variable” mean in algebra?
Isolating the variable means getting the unknown letter alone on one side of the equation so you can read its value directly. You achieve this by applying inverse operations — the opposite of what is being done to the variable — to both sides of the equation equally.
Can I use this worksheet for free?
Yes, completely free. The on-page worksheet is available right here, and the downloadable PDF (with a separate answer key section) is also free. No account or sign-up is required. Click either download button on this page.
What is an inverse operation in algebra?
An inverse operation is the mathematical opposite of another operation. Addition and subtraction are inverses of each other; multiplication and division are inverses of each other. To solve an equation, you apply the inverse of whatever is being done to the variable to “undo” it and isolate x.
Key Takeaways
- A linear equation has a variable with exponent 1 and exactly one solution.
- The four-step method (simplify, move variables left, move constants right, divide) works for every linear equation.
- Always expand brackets before applying other steps.
- The most common error is a sign mistake when moving terms — think “add the opposite to both sides” instead.
- Always check your answer by substituting it back into the original equation.
- This free worksheet has 10 problems from one-step to fraction equations, with a full answer key.
Dr. Irfan Mansuri
Dr. Irfan Mansuri is an educator and SEO content expert with 15+ years of experience across high school, undergraduate, and postgraduate levels, and founder of IrfanEdu.com. I combine deep subject knowledge with proven test-taking strategies to make complex ideas simple. Connect on LinkedIn.
Sources & References
- Khan Academy — Linear Equations: Variables on Both Sides (Review)
- Wikipedia — Linear Equation
- Encyclopaedia Britannica — Algebra
Editorial note: All worked examples and problems in this article were independently verified by Dr. Irfan Mansuri. No statistics or claims have been fabricated. This article was last reviewed on July 11, 2026.