GCSE — mathsFoundation8 min read

Solving Linear Equations

Master solving linear equations for GCSE Maths. Covers one-step, two-step, brackets and unknowns on both sides with worked examples.

Tutor AI Team
Tutor AI Team

Solving linear equations is one of the most fundamental skills in GCSE Maths. A linear equation is an equation where the highest power of the unknown (usually xx) is 1. The goal is always the same: find the value of the unknown that makes the equation true.

Linear equations appear throughout your GCSE course — from simple one-step problems on the Foundation tier right through to forming and solving equations in context on the Higher tier. Mastering this topic gives you a solid foundation for algebra, simultaneous equations, and beyond.

In this guide, you will learn how to solve one-step equations, two-step equations, equations with brackets, and equations with unknowns on both sides. Every technique follows one golden rule: whatever you do to one side, you must do to the other.

Core Concepts

What Is a Linear Equation?

A linear equation is a mathematical statement that two expressions are equal, where the variable appears only to the power of 1. For example:

3x+5=203x + 5 = 20

The left-hand side (LHS) equals the right-hand side (RHS) for exactly one value of xx. Your job is to find that value.

The Balance Method

Think of an equation as a set of balance scales. Both sides must remain equal. If you add 3 to one side, you must add 3 to the other. If you divide one side by 4, you must divide the other by 4 as well.

The operations you use to isolate the unknown are called inverse operations:

  • The inverse of addition is subtraction
  • The inverse of subtraction is addition
  • The inverse of multiplication is division
  • The inverse of division is multiplication

One-Step Equations

These require just one operation to solve.

Example: Solve x+7=12x + 7 = 12

Subtract 7 from both sides: x=12−7=5x = 12 - 7 = 5

Example: Solve 4x=284x = 28

Divide both sides by 4: x=28÷4=7x = 28 \div 4 = 7

Two-Step Equations

These require two inverse operations. The general approach is to deal with addition or subtraction first, then multiplication or division.

Example: Solve 3x+4=193x + 4 = 19

Step 1: Subtract 4 from both sides: 3x=153x = 15

Step 2: Divide both sides by 3: x=5x = 5

Equations with Brackets

When an equation contains brackets, expand the brackets first, then solve as before.

Example: Solve 2(x+3)=142(x + 3) = 14

Expand: 2x+6=142x + 6 = 14

Subtract 6: 2x=82x = 8

Divide by 2: x=4x = 4

Alternatively, you can divide both sides by 2 first: x+3=7x + 3 = 7, then subtract 3: x=4x = 4. Both approaches are valid.

Unknowns on Both Sides

When the unknown appears on both sides, collect the variable terms on one side and the number terms on the other.

Example: Solve 5x+2=3x+105x + 2 = 3x + 10

Subtract 3x3x from both sides: 2x+2=102x + 2 = 10

Subtract 2 from both sides: 2x=82x = 8

Divide by 2: x=4x = 4

Tip: Always move the smaller xx term to avoid negative coefficients where possible.

Strategy Tips

Tip 1: Always Check Your Answer

Substitute your answer back into the original equation. If both sides are equal, your solution is correct. For 5x+2=3x+105x + 2 = 3x + 10 with x=4x = 4: LHS =5(4)+2=22= 5(4) + 2 = 22, RHS =3(4)+10=22= 3(4) + 10 = 22 ✓

Tip 2: Deal with Fractions Early

If the equation contains fractions, multiply every term by the lowest common denominator (LCD) to clear them. For example, if you see x3+x4=7\frac{x}{3} + \frac{x}{4} = 7, multiply everything by 12.

Tip 3: Keep Your Working Neat

Write each step on a new line. Examiners award method marks for clear, logical working. Even if your final answer is wrong, you can still pick up marks for correct steps.

Tip 4: Watch the Signs

When subtracting a negative or expanding brackets with a negative sign outside, be extra careful. For example, −(x−3)=−x+3-(x - 3) = -x + 3, not −x−3-x - 3.

Tip 5: Form Your Own Equations

In context questions, define your variable clearly (e.g., "Let xx be the number of sweets"), write the equation from the information given, solve it, then answer the question in words.

Worked Example: Example 1

Problem

Solve 7x−3=257x - 3 = 25

Solution

7x−3=257x - 3 = 25

Add 3 to both sides:

7x=287x = 28

Divide both sides by 7:

x=4x = 4

Check: 7(4)−3=28−3=257(4) - 3 = 28 - 3 = 25 ✓

Worked Example: Example 2

Problem

Solve 3(2x−1)=4x+73(2x - 1) = 4x + 7

Solution

Expand the bracket:

6x−3=4x+76x - 3 = 4x + 7

Subtract 4x4x from both sides:

2x−3=72x - 3 = 7

Add 3 to both sides:

2x=102x = 10

Divide both sides by 2:

x=5x = 5

Check: LHS =3(2×5−1)=3(9)=27= 3(2 \times 5 - 1) = 3(9) = 27. RHS =4(5)+7=27= 4(5) + 7 = 27 ✓

Worked Example: Example 3

Problem

The perimeter of a rectangle is 38 cm. The length is xx cm and the width is (x−5)(x - 5) cm. Find the dimensions of the rectangle.

Solution

Perimeter =2(length+width)= 2(\text{length} + \text{width})

2(x+x−5)=382(x + x - 5) = 38

2(2x−5)=382(2x - 5) = 38

Divide both sides by 2:

2x−5=192x - 5 = 19

Add 5:

2x=242x = 24

Divide by 2:

x=12x = 12

The length is 1212 cm and the width is 12−5=712 - 5 = 7 cm.

Check: 2(12+7)=2×19=382(12 + 7) = 2 \times 19 = 38 cm ✓

Worked Example: Example 4

Problem

Solve x+13=x−22\frac{x + 1}{3} = \frac{x - 2}{2}

Solution

Multiply both sides by 6 (the LCD of 3 and 2):

2(x+1)=3(x−2)2(x + 1) = 3(x - 2)

Expand:

2x+2=3x−62x + 2 = 3x - 6

Subtract 2x2x from both sides:

2=x−62 = x - 6

Add 6:

x=8x = 8

Check: LHS =8+13=93=3= \frac{8+1}{3} = \frac{9}{3} = 3. RHS =8−22=62=3= \frac{8-2}{2} = \frac{6}{2} = 3 ✓

Practice Problems

  1. Problem 1

    Solve 4x+9=334x + 9 = 33

    Problem 2

    Solve 5(x−2)=3x+65(x - 2) = 3x + 6

    Problem 3

    Solve 8x−5=3x+208x - 5 = 3x + 20

    Problem 4

    Solve 2x+15=3\frac{2x + 1}{5} = 3

    Problem 5

    Three consecutive integers add up to 51. By forming an equation, find the three integers.

    Problem 6

    Solve 4(2x+3)=2(3x−1)+204(2x + 3) = 2(3x - 1) + 20

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Common Mistakes

  • Forgetting to apply operations to both sides. If you subtract 5 from the left side, you must subtract 5 from the right side too.
  • Errors when expanding brackets with negatives. Remember: −2(x−4)=−2x+8-2(x - 4) = -2x + 8, not −2x−8-2x - 8.
  • Dividing only one term. When dividing both sides by a number, every term on each side must be divided. For instance, 3x+63=x+2\frac{3x + 6}{3} = x + 2, not x+6x + 6.
  • Not checking the answer. Always substitute your value back in to verify. This takes 10 seconds and can save you marks.
  • Losing the variable. When collecting terms, students sometimes subtract the wrong xx term, ending up with 0x0x on both sides. Always subtract the smaller coefficient from the larger.

Frequently Asked Questions

What does "solve" mean in maths?

To solve an equation means to find the value of the unknown variable that makes the equation true. For a linear equation, there is exactly one solution.

Do I always have to show my working?

Yes. GCSE mark schemes award method marks for each correct step. Even if you can do it in your head, write the steps down. If your final answer is wrong, you may still get marks for correct working.

What if I get a negative answer?

That is perfectly fine. Many equations have negative solutions. For example, 2x+10=42x + 10 = 4 gives x=−3x = -3. Always check by substituting back in.

What if $x$ is a fraction or decimal?

Linear equations can have fractional or decimal solutions. For example, 2x+1=62x + 1 = 6 gives x=2.5x = 2.5 or x=52x = \frac{5}{2}. Either form is usually acceptable unless the question specifies otherwise.

How do I know which side to collect the $x$ terms on?

Collect the xx terms on the side where the coefficient is larger. This avoids working with negative coefficients, which reduces the chance of errors.

Key Takeaways

  • ✓

    One golden rule. Whatever you do to one side of the equation, you must do to the other side.

  • ✓

    Inverse operations are your main tool. Use addition/subtraction and multiplication/division to isolate the unknown step by step.

  • ✓

    Expand brackets first. If the equation contains brackets, expand them before collecting like terms.

  • ✓

    Collect unknowns on one side. When xx appears on both sides, move the xx terms to one side and the numbers to the other.

  • ✓

    Always verify your solution. Substitute your answer back into the original equation to check it works.

  • ✓

    Show every step. Clear, logical working earns method marks and helps you spot errors.

Tags:
#gcse#maths#algebra#equations

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