Solve: \(x + y = 10\) \(x = 4\)
Substitute x = 4 into the first equation.
Substitute \(x = 4\): \(4 + y = 10\), so \(y = 6\).
Simultaneous Equations
Simultaneous equations are solved by finding values that satisfy two equations at the same time. Methods such as substitution and elimination are commonly used. These equations can also be understood visually using linear graphs, where the solution is the point of intersection.
Overview
Simultaneous equations are two equations with the same variables.
You solve them together to find values that make both equations true at the same time.
\( \begin{cases} x + y = 10 \\ x - y = 2 \end{cases} \)
The solution is the pair of values for \(x\) and \(y\) that satisfies both equations.
What you should understand after this topic
- Understand what simultaneous equations mean
- Solve simultaneous equations by elimination
- Solve simultaneous equations by substitution
- Check answers correctly
- Understand how solutions connect to graph intersections
Key Definitions
Simultaneous Equations
Two or more equations solved together.
Solution
The values that make all equations true at the same time.
Elimination
A method where one variable is removed by adding or subtracting equations.
Substitution
A method where one equation is used inside the other.
Intersection
The point where two graphs meet.
Linear Equation
An equation with variables of power 1, such as \(2x + y = 7\).
Key Rules
Same variables
The equations must involve the same variables, usually \(x\) and \(y\).
Eliminate one variable
Try to remove either \(x\) or \(y\) first.
Substitute back
After finding one variable, use it to find the other.
Check both equations
Your answer must work in both equations.
Quick Method Check
Elimination works well
When coefficients already match or are easy to match.
Substitution works well
When one equation is already rearranged, like \( y = 2x + 1 \).
Final answer
Usually written as \(x = \dots,\ y = \dots\).
Graph meaning
The solution is where the two lines intersect.
How to Solve
What does “simultaneous” mean?
Simultaneous equations are solved together. You are looking for one pair of values that satisfies both equations.
\( x + y = 8 \)
\( x - y = 4 \)
The solution must work in both equations at the same time.
Method 1: Elimination
Elimination removes one variable by adding or subtracting the equations.
- Make the coefficients of one variable the same.
- Add or subtract the equations to eliminate that variable.
- Solve the resulting equation.
- Substitute back to find the second variable.
Elimination with scaling
\( 2x + y = 11 \)
\( x - y = 1 \)
Here, \(+y\) and \(-y\) already cancel.
If they did not match, multiply one or both equations first.
Exam tip: Always align coefficients before eliminating.
Method 2: Substitution
Substitution replaces one variable using another equation.
- Rearrange one equation to make a variable the subject.
- Substitute into the other equation.
- Solve the resulting equation.
- Substitute back to find the second variable.
Choosing the best method
Use elimination
When coefficients are equal or easy to match.
Use substitution
When one equation is already rearranged.
Connection to graphs
Each equation represents a straight line. The solution is where the two lines intersect.
Key idea: Intersection = solution.
Exam thinking
Always substitute your answers back to check.
Take care with negative signs when eliminating.
Exam tip: Show each step clearly for full marks.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics.
Edexcel
Solve the simultaneous equations: \( x + y = 10 \) and \( x - y = 2 \).
Edexcel
Solve the simultaneous equations: \( 2x + y = 7 \) and \( x + y = 5 \).
Edexcel
Solve the simultaneous equations: \( 3x + 2y = 12 \) and \( x + 2y = 8 \).
Edexcel
Solve the simultaneous equations: \( 4x - y = 5 \) and \( 2x + y = 11 \).
Edexcel
Solve the simultaneous equations and give your answers for x and y: 5x + y = 16 and 2x + y = 7.
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on substitution, elimination, and contextual applications.
AQA
Solve the simultaneous equations: \( y = 2x + 1 \) and \( y = x + 5 \).
AQA
Solve the simultaneous equations: \( y = 3x - 4 \) and \( x + y = 10 \).
AQA
Solve the simultaneous equations: \( 2x + 3y = 13 \) and \( x - y = 1 \).
AQA
Two pens and three pencils cost £2.10. Four pens and one pencil cost £2.30. Form and solve a pair of simultaneous equations to find the cost of each item.
AQA
The sum of two numbers is 20 and their difference is 4. Form and solve a pair of simultaneous equations to find the two numbers.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising graphical interpretation and higher-level problem solving.
OCR
Solve the simultaneous equations: \( y = x + 2 \) and \( y = -x + 8 \).
OCR
The lines y = 2x + 1 and y = -x + 7 intersect. Find the coordinates of the point of intersection.
OCR
Solve the simultaneous equations: \( x^2 + y^2 = 25 \) and \( y = x + 1 \).
OCR
Two cinema tickets and three drinks cost £19. Three cinema tickets and two drinks cost £21. Form and solve a pair of simultaneous equations to find the cost of one ticket and one drink.
OCR
Explain how the graphical method can be used to solve simultaneous equations.
Exam Checklist
Step 1
Decide whether elimination or substitution is the better method.
Step 2
Find one variable first.
Step 3
Substitute back to find the second variable.
Step 4
Check both values in both equations.
Most common exam mistakes
Sign error
Adding or subtracting equations incorrectly.
Incomplete answer
Finding only \(x\) or only \(y\).
Substitution error
Replacing the wrong expression or missing brackets.
No checking
Not confirming the answer works in both equations.
Common Mistakes
These are common mistakes students make when solving simultaneous equations in GCSE Maths.
Stopping after finding one variable
Incorrect
A student finds one value but does not continue.
Correct
You must find both variables. After solving for one, substitute it back into an equation to find the other.
Sign errors during elimination
Incorrect
A student makes mistakes when adding or subtracting equations.
Correct
Be careful with signs when eliminating variables. Double-check whether you should add or subtract the equations.
Incorrect substitution
Incorrect
A student substitutes a value incorrectly into the equation.
Correct
Replace the variable carefully and use brackets where needed, especially for negative values.
Not checking the solution
Incorrect
A student assumes the answer is correct without verification.
Correct
Substitute both values back into both original equations to confirm they satisfy each one.
Mixing methods
Incorrect
A student switches between elimination and substitution without a clear plan.
Correct
Choose one method and follow it step by step to avoid confusion.
Try It Yourself
Practise solving simultaneous equations algebraically and graphically.
Foundation
Foundation Practice
Solve simultaneous equations using substitution.
Solve: \(x + y = 8\) \(x = 3\) Find y.
Substitute x into the equation.
\(3 + y = 8\), so \(y = 5\).
Solve: \(x + y = 9\) \(y = 2\)
Substitute y = 2.
\(x + 2 = 9\), so \(x = 7\).
Solve: \(x + y = 12\) \(y = 5\) Find x.
Substitute y = 5.
\(x + 5 = 12\), so \(x = 7\).
Solve: \(x + y = 10\) \(x - y = 2\)
Add the two equations.
Add equations: \(2x = 12\), so \(x = 6\). Substitute: \(6 + y = 10\), so \(y = 4\).
Solve: \(x + y = 14\) \(x - y = 4\) Give x and y as x,y.
Add the equations.
Add: \(2x = 18\), so \(x = 9\). Substitute: \(9 + y = 14\), so \(y = 5\).
A student solves: \(x + y = 10\) \(x = 4\) and gets \(y = 14\). What is the mistake?
Check the calculation 10 - 4.
They should subtract: \(10 - 4 = 6\), not add.
Solve: \(x + y = 6\) \(x = 2\) Find y.
Substitute x = 2.
\(2 + y = 6\), so \(y = 4\).
Solve: \(x + y = 5\) \(y = 1\)
Substitute y = 1.
\(x + 1 = 5\), so \(x = 4\).
Solve: \(x + y = 20\) \(x = 8\) Find y.
Substitute x = 8.
\(8 + y = 20\), so \(y = 12\).
Higher
Higher Practice
Solve simultaneous equations using substitution and elimination.
Solve: \(x + y = 10\) \(2x + y = 14\)
Subtract the first equation from the second.
Subtract: \(x = 4\). Substitute: \(4 + y = 10\), so \(y = 6\).
Solve: \(2x + y = 11\) \(x + y = 7\) Give x,y.
Subtract the equations.
Subtract: \(x = 4\). Substitute: \(4 + y = 7\), so \(y = 3\).
Solve: \(y = x + 2\) \(x + y = 8\)
Substitute y into the second equation.
Substitute: \(x + (x + 2) = 8\) → \(2x = 6\) → \(x = 3\), so \(y = 5\).
Solve: \(y = 2x\) \(x + y = 12\) Give x,y.
Substitute y into the second equation.
\(x + 2x = 12\) → \(3x = 12\) → \(x = 4\), so \(y = 8\).
Solve: \(3x + y = 13\) \(x + y = 7\)
Subtract the equations.
Subtract: \(2x = 6\), so \(x = 3\). Substitute: \(3 + y = 7\), so \(y = 4\).
Solve: \(2x + y = 10\) \(x + y = 6\) Give x,y.
Subtract the equations.
Subtract: \(x = 4\). Substitute: \(4 + y = 6\), so \(y = 2\).
Solve: \(y = 3x\) \(x + y = 16\)
Substitute y = 3x.
\(x + 3x = 16\) → \(4x = 16\) → \(x = 4\), so \(y = 12\).
Solve: \(y = x - 1\) \(x + y = 9\) Give x,y.
Substitute y into the second equation.
\(x + (x - 1) = 9\) → \(2x = 10\) → \(x = 5\), so \(y = 4\).
A student solves: \(x + y = 10\) \(2x + y = 14\) and gets \(x = 10\). What mistake did they make?
Check how they eliminated y.
They should subtract equations to eliminate y, not add incorrectly.
Solve: \(3x + y = 17\) \(x + y = 9\) Give x,y.
Subtract the equations.
Subtract: \(2x = 8\) → \(x = 4\). Substitute: \(4 + y = 9\), so \(y = 5\).
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What are simultaneous equations?
Two equations solved together to find common solutions.
What methods can I use?
Substitution, elimination or graphing.
How do I know the answer is correct?
It must satisfy both equations.