What you'll learn
Simultaneous equations are two equations with two unknowns, solved together to find values that satisfy both. In this guide you will learn how to solve linear simultaneous equations by elimination and by substitution, how to interpret the solution graphically, and how to set up simultaneous equations from word problems. These methods are essential for many GCSE algebra and problem-solving questions.
Key terms and definitions
Simultaneous equations — two (or more) equations that share the same unknowns and are true at the same time.
Elimination — adding or subtracting equations to remove one variable.
Substitution — rearranging one equation and substituting it into the other.
Solution — the pair of values (x, y) that satisfies both equations.
Coefficient — the number multiplying a variable.
Core concepts
What simultaneous equations mean
Two linear simultaneous equations each describe a straight line. The solution is the pair of values (x and y) that makes both equations true — graphically, the point where the two lines cross. There are two main algebraic methods: elimination and substitution.
Solving by elimination
In elimination, you make the coefficients of one variable match, then add or subtract the equations to remove that variable:
- Multiply one or both equations so one variable has equal coefficients.
- Add the equations if the signs are opposite, or subtract if they are the same, to eliminate that variable.
- Solve for the remaining variable.
- Substitute back to find the other variable.
Solving by substitution
In substitution, you rearrange one equation to make one variable the subject, then substitute that expression into the other equation:
- Rearrange one equation (e.g. y = …).
- Substitute into the other equation, giving one equation in one unknown.
- Solve it, then substitute back to find the other variable.
Substitution is often easiest when one equation already has a variable on its own.
Checking your answer
Always substitute both values back into the original equations to check they work. This catches arithmetic and sign errors.
Setting up from word problems
Many problems give two pieces of information that translate into two equations — for example, total cost and total quantity. Define your variables clearly, write the two equations, then solve as usual.
Worked examples
Example 1: Elimination
Solve 3x + y = 11 and 2x + y = 8.
Subtract the equations (y eliminated): (3x − 2x) = 11 − 8, so x = 3. Substitute: 3(3) + y = 11, so y = 2. x = 3, y = 2.
Example 2: Substitution
Solve y = 2x + 1 and 3x + y = 16.
Substitute: 3x + (2x + 1) = 16 → 5x + 1 = 16 → 5x = 15 → x = 3. Then y = 2(3) + 1 = 7. x = 3, y = 7.
Example 3: Multiplying first
Solve 2x + 3y = 13 and x − y = 1.
Multiply the second by 3: 3x − 3y = 3. Add to the first: 5x = 16... instead, multiply second by 2: 2x − 2y = 2; subtract from first: 5y = 11, y = 2.2 — better: from x − y = 1, x = y + 1; substitute: 2(y + 1) + 3y = 13 → 5y + 2 = 13 → y = 2.2, x = 3.2. x = 3.2, y = 2.2.
Common mistakes and how to avoid them
Adding when you should subtract (or vice versa). Add if the matching coefficients have opposite signs; subtract if the same.
Forgetting to multiply the whole equation. When scaling an equation, multiply every term.
Only finding one variable. Substitute back to find the second value.
Sign errors in substitution. Use brackets when substituting an expression.
Not checking. Put both values into both original equations.
Exam technique for Simultaneous Equations
Choose a method — elimination if coefficients match easily, substitution if one variable is already isolated.
Match coefficients carefully when using elimination.
Use brackets when substituting to avoid sign mistakes.
Find both variables and write the full solution.
Check by substitution into both equations.
Quick revision summary
Simultaneous equations are solved together to find the (x, y) pair that satisfies both — graphically, the point where the lines cross. By elimination, scale the equations so one variable has matching coefficients, then add (opposite signs) or subtract (same signs) to remove it, solve for the remaining variable, and substitute back. By substitution, rearrange one equation to make a variable the subject, substitute it into the other (using brackets to avoid sign errors), solve the single-variable equation, then back-substitute. Always find both variables and check by substituting into both original equations. For word problems, define variables clearly and translate the two facts into two equations. The common pitfalls are adding/subtracting incorrectly, forgetting to multiply every term when scaling, stopping after one variable, and sign slips. Pick the easier method, work carefully with signs and brackets, give the full solution, and verify it satisfies both equations.