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Simultaneous Equations GCSE Maths

Simultaneous equations are an important GCSE algebra topic. The aim is to find values that satisfy two equations at the same time.

Students usually solve simultaneous equations using either the elimination method or the substitution method.

Video explanation

A short Worthing Maths Tutor video explanation for simultaneous equations GCSE maths can be embedded here later to improve student engagement and time on page.

What are simultaneous equations?

Simultaneous equations are equations solved together because both equations are true at the same time.

For example:

x + y = 10
x - y = 2

We need to find values of x and y that work in both equations.

Method 1: Elimination

Elimination means removing one variable by adding or subtracting the equations.

Example 1: Solving by elimination

x + y = 10
x - y = 2

Step 1: Add the equations together

2x = 12

Step 2: Divide by 2

x = 6

Step 3: Substitute into one equation

6 + y = 10
y = 4

Answer:

x = 6, y = 4

Example where coefficients must match first

Example 2: Matching coefficients

2x + y = 11
3x - y = 9

The y terms already have opposite signs, so add the equations.

5x = 20
x = 4

Substitute into the first equation:

2(4) + y = 11
8 + y = 11
y = 3

Answer:

x = 4, y = 3

Method 2: Substitution

Substitution means rearranging one equation and replacing the variable in the other equation.

Example 3: Solving by substitution

y = x + 2
2x + y = 11

Step 1: Substitute y = x + 2 into the second equation

2x + (x + 2) = 11

Step 2: Simplify

3x + 2 = 11
3x = 9
x = 3

Step 3: Find y

y = 3 + 2 = 5

Answer:

x = 3, y = 5
Exam tip: If one equation is already rearranged, substitution is often the quickest method.

How to check answers

Substitute your answers back into both original equations.

If both equations work correctly, your solution is correct.

Common mistakes in simultaneous equations

  • Adding equations incorrectly.
  • Sign mistakes with negative numbers.
  • Forgetting to substitute back to find the second variable.
  • Choosing elimination before matching coefficients properly.
  • Arithmetic mistakes during substitution.
Common mistake: Some students find x correctly but forget to calculate y afterwards. Always give values for both variables unless the question says otherwise.

Foundation vs Higher GCSE

Foundation GCSE questions usually involve linear simultaneous equations with integer solutions.

Higher GCSE may include:

  • harder elimination problems
  • equations with brackets
  • fractional coefficients
  • quadratic simultaneous equations
  • graphical solutions

Practice questions

  1. Solve:
    x + y = 7
    x - y = 1
  2. Solve:
    2x + y = 9
    x + y = 6
  3. Solve:
    y = x + 1
    2x + y = 10
  4. Solve:
    3x + y = 14
    2x - y = 1

Answers

  1. x = 4, y = 3
  2. x = 3, y = 3
  3. x = 3, y = 4
  4. x = 3, y = 5

Simultaneous equations FAQ

What are simultaneous equations?

Simultaneous equations are two or more equations solved together to find values that satisfy all equations at the same time.

What methods are used to solve simultaneous equations?

The main GCSE methods are elimination and substitution. Higher GCSE may also include graphical methods and nonlinear simultaneous equations.

Are simultaneous equations on Foundation GCSE?

Yes. Simultaneous equations appear on both Foundation and Higher GCSE papers, although Higher questions are usually more difficult.

How can I get better at simultaneous equations?

Practise identifying whether elimination or substitution is easier, keep work organised carefully, and check answers by substituting them back into both equations.

Related GCSE algebra topics

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