[Leejohnston]

Equation Solving Basics — Step-by-Step for Beginners

Solving equations is about finding the value of the unknown number (usually x). 
The goal is simple:

Get x by itself.

Here is the beginner-friendly guide.

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1. What Is an Equation?

An equation has:
• two sides 
• an equals sign 
• an unknown (x, y, etc.)

Examples:
• x + 5 = 12 
• 4x = 20 
• 3x – 2 = 13 

An equation is true only when x has the correct value.

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2. Opposite Operations

To solve equations, you “undo” what is happening.

Use the opposite operation:

Add ↔ Subtract 
× ↔ ÷ 

Examples:
• If the equation adds 6 → subtract 6 
• If it multiplies by 4 → divide by 4 

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3. Solving One-Step Equations

Example 1: 
x + 7 = 15 
→ subtract 7 
x = 8

Example 2: 
4x = 28 
→ divide by 4 
x = 7

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4. Solving Two-Step Equations

Example: 
3x + 5 = 20

Step 1: subtract 5 
3x = 15

Step 2: divide by 3 
x = 5

Another example: 
2x – 4 = 10 
Add 4 → 2x = 14 
Divide by 2 → x = 7

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5. Solving Equations with Brackets

Expand the bracket first.

Example: 
2(x + 3) = 14 
→ 2x + 6 = 14 
→ subtract 6 → 2x = 8 
→ divide 2 → x = 4

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6. Solving Equations with Fractions

Example: 
x/3 = 7 
Multiply both sides by 3 → x = 21

Example: 
(x + 1)/5 = 4 
Multiply both sides by 5 → x + 1 = 20 
Then subtract 1 → x = 19

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7. Checking Your Answer

Always substitute back in:

Example: 
Equation: 3x + 1 = 10 
Your answer: x = 3 
Check: 3(3) + 1 = 10 ✔

Checking helps catch mistakes — especially on exams.

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8. Common GCSE Mistakes

❌ Forgetting to do the same thing to both sides 
❌ Expanding brackets incorrectly 
❌ Adding instead of subtracting 
❌ Rushing negative numbers 
❌ Not checking the answer 

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Summary

To solve equations:
• Isolate x 
• Use opposite operations 
• Work step-by-step 
• Expand brackets if needed 
• Check your answer 

Master these basics and all algebra becomes easier — including GCSE problem-solving.