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Study Guide: How to Solve: Linear Equations
Source: https://www.fatskills.com/k-12-assessment-tests/chapter/how-to-solve-linear-equations

How to Solve: Linear Equations

By Fatskills Exam Guides Team — the exam nerds behind 28,500+ quizzes and 2.1M practice questions across 500+ global exams.

⏱️ ~4 min read

How to Solve: Linear Equations

For Students Who Want to Ace Their Exam & Teachers Who Need a Ready-to-Record Script

Introduction

"Master linear equations, and you’ll unlock everything from budgeting your allowance to cracking algebra exams—this is the skill that turns ‘I don’t get it’ into ‘I’ve got this.’"

What You Need To Know First

Before diving into linear equations, ensure you understand: 1. Variables and constants – Letters (like x) represent unknowns; numbers (like 5) are fixed. 2. Inverse operations – Addition undoes subtraction, multiplication undoes division. 3. Order of operations (PEMDAS/BODMAS) – Parentheses/Brackets first, then Exponents/Orders, then Multiply/Divide, finally Add/Subtract.

Key Vocabulary

Term Plain-English Definition Quick Example
Linear equation An equation where the highest power of x is 1. 2x + 3 = 7
Solution The value of x that makes the equation true. For 2x + 3 = 7, x = 2 is the solution.
Isolate Move all terms with x to one side, constants to the other. x + 5 = 9 → x = 9 – 5
Coefficient The number multiplied by x. In 4x, the coefficient is 4.
Like terms Terms with the same variable (or no variable). 3x and 5x are like terms; 3x and 5 are not.
Distributive property Multiply a term outside parentheses by each term inside. 2(x + 3) = 2x + 6

Formulas To Know

1. Standard Form of a Linear Equation

Formula: ax + b = c - a = coefficient of x (MEMORISE THIS) - b = constant term (MEMORISE THIS) - c = constant term on the other side (MEMORISE THIS)

When to use: Any linear equation can be rewritten in this form.

2. Solving for x (General Method)

Formula: x = (c – b) / a - Derived from ax + b = c (MEMORISE THIS as a shortcut for simple equations).

When to use: After isolating x in one-step or two-step equations.

Step-by-Step Method

Goal: Solve for x in any linear equation.

Step 1: Simplify Both Sides

  • Remove parentheses using the distributive property.
  • Example: 3(x + 2) = 12 → 3x + 6 = 12
  • Combine like terms on each side.
  • Example: 2x + 3x – 5 = 10 → 5x – 5 = 10

Step 2: Move Constants to One Side

  • Use inverse operations to move constants (numbers without x) to the opposite side of x.
  • Example: 5x – 5 = 10
    • Add 5 to both sides: 5x = 15

Step 3: Move Coefficients to the Other Side

  • Divide both sides by the coefficient of x to isolate x.
  • Example: 5x = 15
    • Divide by 5: x = 3

Step 4: Check Your Solution

  • Substitute x back into the original equation to verify.
  • Example: For 3(x + 2) = 12, substitute x = 2:
    • 3(2 + 2) = 3(4) = 12 ✓

Worked Example Using the Steps

Equation: 4(x – 3) + 2 = 18

Step Action Equation After Step
1 Distribute the 4: 4 × x and 4 × –3 4x – 12 + 2 = 18
1 Combine like terms: –12 + 2 4x – 10 = 18
2 Add 10 to both sides 4x = 28
3 Divide both sides by 4 x = 7
4 Check: 4(7 – 3) + 2 = 4(4) + 2 = 18 ✓ Solution: x = 7

Worked Examples

Example 1 – Basic (One-Step Equation)

Equation: x + 9 = 15

Step Action Working
1 Subtract 9 from both sides x = 15 – 9
2 Simplify x = 6
3 Check: 6 + 9 = 15 ✓ Solution: x = 6

What we did and why: We used the inverse operation (subtraction) to isolate x. Since x was being added to 9, we subtracted 9 from both sides to balance the equation.

Example 2 – Medium (Two-Step Equation with Fractions)

Equation: 3x – 4 = 11

Step Action Working
1 Add 4 to both sides 3x = 11 + 4
2 Simplify 3x = 15
3 Divide by 3 x = 15 ÷ 3
4 Simplify x = 5
5 Check: 3(5) – 4 = 15 – 4 = 11 ✓ Solution: x = 5

What we did and why: First, we moved the constant (–4) to the other side using addition. Then, we divided by the coefficient (3) to solve for x. Always do addition/subtraction before multiplication/division.

Example 3 – Exam Style (Disguised Equation)

Equation: 2(3x + 1) – 5x = 7

Step Action Working
1 Distribute the 2 6x + 2 – 5x = 7
2 Combine like terms (6x – 5x) x + 2 = 7
3 Subtract 2 from both sides x = 7 – 2
4 Simplify x = 5
5 Check: 2(3(5) + 1) – 5(5) = 2(16) – 25 = 32 – 25 = 7 ✓ Solution: x = 5

What we did and why: We simplified the equation by distributing first, then combining like terms. This made it a simple one-step equation. Always simplify before solving!

Common Mistakes

Mistake Why It Happens Correct Approach
Forgetting to distribute Rushing and skipping the distributive property. Always multiply the term outside parentheses by every term inside.
Sign errors Misplacing a negative sign when moving terms. Write every step clearly. Example: –3x + 5 = 2 → –3x = –3 (not 3x = –3).
Dividing before adding Trying to divide first in a two-step equation. Always move constants first, then coefficients. Example: 3x + 2 = 8 → 3x = 6 → x = 2.
Combining unlike terms Adding x terms to constants. Example: 3x + 2 = 5x → 5x = 2 (wrong). Only combine terms with the same variable. 3x + 2 = 5x → 2 = 2x.
Not checking the solution Assuming the answer is correct without verifying. Always plug x back into the original equation to confirm.

Exam Traps

Trap How to Spot It How to Avoid It
Hidden parentheses Equations like 2(x + 3) = 10 look simple but require distribution. Always look for parentheses—distribute first!
Fractions or decimals Equations like 0.5x + 2 = 4 or x/3 = 5. Multiply both sides by the denominator (or 10 for decimals) to eliminate fractions/decimals. Example: x/3 = 5 → x = 15.
Variables on both sides Equations like 3x + 2 = 2x + 7. Move all x terms to one side and constants to the other. Example: 3x – 2x = 7 – 2 → x = 5.

1-Minute Recap

"Alright, let’s lock this in—tonight or right before your exam. Here’s the game plan for linear equations:

  1. Simplify first: Distribute, then combine like terms. No shortcuts here.
  2. Move constants: Get all the numbers without x to one side using inverse operations.
  3. Isolate x: Divide by the coefficient to solve for x.
  4. Check your answer: Plug it back into the original equation. If it works, you’re golden.

Remember the traps: watch for parentheses, fractions, and variables on both sides. And if you’re stuck, write every step—no skipping! You’ve got this. Now go ace that exam!


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