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Linear EquationFREEWorksheet Generator(Grades 6-9)

Linear Equation worksheets with step-by-step answer keys. Generate unlimited unique practice sheets with fresh problems for Grades 6–9.

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Set A • 10 Qs

PanMaths.comLinear Equation Worksheets • Set A (Medium)

Name: __________________________

Date: _____________ Period: ____

Score: _____ / 10

5x+2=12

Workspace:

6x−9=21

Workspace:

6x+4=58

Workspace:

4x+11=47

Workspace:

5x−14=6

Workspace:

3x+8=26

Workspace:

3x+11=23

Workspace:

4x+10=18

Workspace:

2x+13=21

Workspace:

10.

3x+7=13

Workspace:

PanMaths.com • Free Math Worksheet Generator • Set A (Medium)Page 1 of 2

PanMaths.comAnswer Key & Worked Solutions • Set A (Medium)

Answer Key

Total Problems: 10

5x+2=12Ans: x = 2

1. Isolate Variable Term: Subtract 2 from both sides → 5x = 10

2. Divide by Coefficient: x = 10 / 5 → x = 2

3. Proof Check: 5(2) + 2 = 10 + 2 = 12 ✅

6x−9=21Ans: x = 5

1. Isolate Variable Term: Add 9 to both sides → 6x = 30

2. Divide by Coefficient: x = 30 / 6 → x = 5

3. Proof Check: 6(5) − 9 = 30 − 9 = 21 ✅

6x+4=58Ans: x = 9

1. Isolate Variable Term: Subtract 4 from both sides → 6x = 54

2. Divide by Coefficient: x = 54 / 6 → x = 9

3. Proof Check: 6(9) + 4 = 54 + 4 = 58 ✅

4x+11=47Ans: x = 9

1. Isolate Variable Term: Subtract 11 from both sides → 4x = 36

2. Divide by Coefficient: x = 36 / 4 → x = 9

3. Proof Check: 4(9) + 11 = 36 + 11 = 47 ✅

5x−14=6Ans: x = 4

1. Isolate Variable Term: Add 14 to both sides → 5x = 20

2. Divide by Coefficient: x = 20 / 5 → x = 4

3. Proof Check: 5(4) − 14 = 20 − 14 = 6 ✅

3x+8=26Ans: x = 6

1. Isolate Variable Term: Subtract 8 from both sides → 3x = 18

2. Divide by Coefficient: x = 18 / 3 → x = 6

3. Proof Check: 3(6) + 8 = 18 + 8 = 26 ✅

3x+11=23Ans: x = 4

1. Isolate Variable Term: Subtract 11 from both sides → 3x = 12

2. Divide by Coefficient: x = 12 / 3 → x = 4

3. Proof Check: 3(4) + 11 = 12 + 11 = 23 ✅

4x+10=18Ans: x = 2

1. Isolate Variable Term: Subtract 10 from both sides → 4x = 8

2. Divide by Coefficient: x = 8 / 4 → x = 2

3. Proof Check: 4(2) + 10 = 8 + 10 = 18 ✅

2x+13=21Ans: x = 4

1. Isolate Variable Term: Subtract 13 from both sides → 2x = 8

2. Divide by Coefficient: x = 8 / 2 → x = 4

3. Proof Check: 2(4) + 13 = 8 + 13 = 21 ✅

10.

3x+7=13Ans: x = 2

1. Isolate Variable Term: Subtract 7 from both sides → 3x = 6

2. Divide by Coefficient: x = 6 / 3 → x = 2

3. Proof Check: 3(2) + 7 = 6 + 7 = 13 ✅

PanMaths.com • Answer Key & Worked SolutionsPage 2 of 2

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Worksheet Info

Categoryalgebra

Grade LevelGrades 6-9

Difficultymedium

Time10-15 min

Skills Practiced

✓ Linear Equation fluency
✓ Mental math speed
✓ Step-by-step problem solving

Key Strategies

SADMEP Order: Undo addition/subtraction before multiplication/division.
The Golden Scale Law: Always perform identical operations to both sides of the equals sign.
Substitution Check: Plug the calculated x value back into the original equation to verify.

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Strategies

SADMEP Order: Undo addition/subtraction before multiplication/division.
The Golden Scale Law: Always perform identical operations to both sides of the equals sign.
Substitution Check: Plug the calculated x value back into the original equation to verify.

Mastering Linear Equation: Core Concepts & Mental Models

Linear Equation worksheets with step-by-step answer keys. Generate unlimited unique practice sheets with fresh problems for Grades 6–9.

💡 Essential Strategies for Linear Equation

SADMEP Order

Undo addition/subtraction before multiplication/division.

The Golden Scale Law

Always perform identical operations to both sides of the equals sign.

Substitution Check

Plug the calculated x value back into the original equation to verify.

⚠️ Common Mistakes & Teacher Fixes

⚠️Applying Operations to Only One Side of the Equals Sign

Why students stumble: Students subtract or divide a number on the left side but forget to apply the exact same operation to the right side.

Teacher Fix: Treat the equals sign as a balanced scale. Whatever operation you apply to one side must be identically applied to the other.

Example: For x + 5 = 12: Subtract 5 from BOTH sides (x + 5 − 5 = 12 − 5 → x = 7).

⚠️Dividing Before Isolating Constant Terms (Order of Operations)

Why students stumble: In 2x + 4 = 14, students divide 14 by 2 first, forgetting that the +4 must also be divided by 2 if dividing early.

Teacher Fix: Follow "SADMEP" (PEMDAS in Reverse): Undo Subtraction and Addition FIRST, then undo Multiplication and Division SECOND.

Example: Correct: First subtract 4 (2x = 10), then divide by 2 (x = 5).

⚠️Mixing Up Signs During Transposition (e.g. x − 7 = 15 → x = 15 − 7)

Why students stumble: Students see the subtraction sign and subtract again instead of applying the inverse operation (addition).

Teacher Fix: Always apply the exact opposite operation: the inverse of −7 is +7.

Example: For x − 7 = 15: Add 7 to both sides → x = 15 + 7 = 22.

❓ Frequently Asked Questions

What is the difference between a 1-step and 2-step linear equation?

A 1-step equation requires only one inverse operation to solve (e.g. x + 5 = 12 → subtract 5). A 2-step equation requires two operations: first adding/subtracting the constant, then multiplying/dividing the coefficient (e.g. 2x + 4 = 14).

Are all solutions on these worksheets clean integers?

Yes. All generated linear equation problem sets are mathematically calibrated to yield clean integer solutions (e.g. x = 5, x = −3), preventing awkward repeating decimal confusion.

Do the answer keys show step-by-step inverse operations?

Yes. Every generated answer key details each inverse operation step, the simplified intermediate equation, and a final proof check.

Can I print equations with variables on both sides?

Yes. Set C defaults to equations with variables on both sides (ax + b = cx + d). You can also customize difficulty in the Studio.

The 3 Core Rules for Solving Linear Equations

Solving for the unknown variable x requires isolating the variable using reversible inverse operations:

1. The Golden Scale Law: Whatever mathematical operation you perform on the left side of the equation (addition, subtraction, multiplication, division), you MUST perform on the right side.
2. SADMEP Order (Reverse PEMDAS): Always eliminate constants via addition or subtraction before dividing by the variable's coefficient.
3. Proof Verification: Substitute your calculated solution x back into the original equation to verify that Left Hand Side = Right Hand Side.

Step-by-Step Breakdown: 2-Step Equations (ax + b = c)

A standard two-step equation requires exactly two inverse moves:

Step	Algebraic Action	Example: 3x + 7 = 22
Step 1	Subtract constant `b` from both sides	3x + 7 − 7 = 22 − 7 → 3x = 15
Step 2	Divide both sides by coefficient `a`	3x / 3 = 15 / 3 → x = 5
Step 3	Proof check substitution	3(5) + 7 = 15 + 7 = 22 ✅

Equations with Variables on Both Sides (ax + b = cx + d)

When variables appear on both sides of the equals sign, gather all variable terms onto the side with the larger coefficient first:

5x + 3 = 2x + 18 → (5x − 2x) = 18 − 3 → 3x = 15 → x = 5

This guarantees that the resulting coefficient remains positive, preventing negative sign calculation errors.

Worksheet Facts by Grade Level

Grade Level	Common Core Standard	Target Operations & Complexity	Recommended Drills
Grade 6	CCSS.MATH.CONTENT.6.EE.B.7	Solve real-world one-step linear equations involving non-negative rational numbers	Set A (Standard 10 Problems)
Grade 7	CCSS.MATH.CONTENT.7.EE.B.4	Solve two-step linear equations (px + q = r) fluently with positive and negative integers	Set B (Two-Step Practice)
Grade 8	CCSS.MATH.CONTENT.8.EE.C.7	Solve linear equations in one variable with variables on both sides and distributive grouping	Set C (Challenging Drills)
High School Algebra 1	CCSS.MATH.CONTENT.HSA.REI.B.3	Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters	Set C & Unlimited Studio custom drills

The 3 Core Rules for Solving Linear Equations

Solving for the unknown variable x requires isolating the variable using reversible inverse operations:

1. The Golden Scale Law: Whatever mathematical operation you perform on the left side of the equation (addition, subtraction, multiplication, division), you MUST perform on the right side.
2. SADMEP Order (Reverse PEMDAS): Always eliminate constants via addition or subtraction before dividing by the variable's coefficient.
3. Proof Verification: Substitute your calculated solution x back into the original equation to verify that Left Hand Side = Right Hand Side.

Step-by-Step Breakdown: 2-Step Equations (ax + b = c)

A standard two-step equation requires exactly two inverse moves:

Step	Algebraic Action	Example: 3x + 7 = 22
Step 1	Subtract constant `b` from both sides	3x + 7 − 7 = 22 − 7 → 3x = 15
Step 2	Divide both sides by coefficient `a`	3x / 3 = 15 / 3 → x = 5
Step 3	Proof check substitution	3(5) + 7 = 15 + 7 = 22 ✅

Equations with Variables on Both Sides (ax + b = cx + d)

When variables appear on both sides of the equals sign, gather all variable terms onto the side with the larger coefficient first:

5x + 3 = 2x + 18 → (5x − 2x) = 18 − 3 → 3x = 15 → x = 5

This guarantees that the resulting coefficient remains positive, preventing negative sign calculation errors.

Benefits of Mastery

⚖️ Core Algebraic Fluency

Builds intuition for equation balance, variable isolation, and inverse operations.

🎯 CCSS Standard Aligned

Directly maps to 6.EE.B.7, 7.EE.B.4, 8.EE.C.7, and HSA-REI.B.3 across Grades 6 to 9.

🧠 Logical Deductive Reasoning

Develops disciplined step-by-step problem-solving habits essential for high school STEM.

✅ Step-by-Step Answer Keys

Every generated PDF includes full intermediate inverse operation steps for instant grading.