Free Simultaneous Equations Solver
Enter two or three equations and get both unknowns, solved step by step. Shows substitution AND elimination side by side so you can follow whichever your exam board prefers — AQA, Edexcel, OCR, Cambridge, or NCERT.
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A worked simultaneous equation — substitution from start to verify
Below is one fully worked example plus a short primer so you can see exactly how our AI reasons through a problem.
Example Problem
DEMO- 1
Label the equations
Numbering makes the working easier to read when you substitute back.
- 2
Solve (2) for x (substitution method)
The simpler equation is (2) — rearrange it to make x the subject.
- 3
Substitute into (1)
Replace x in the first equation with (y + 1).
- 4
Expand and solve for y
Collect y terms, subtract 2 from both sides, divide by 5.
- 5
Back-substitute to find x
Use the rearranged (2) to get x once y is known.
- 6
Verify in both equations
(1): 2(3) + 3(2) = 12 ✓. (2): 3 − 2 = 1 ✓. Solution is unique.
Final Answer
How to solve simultaneous equations — substitution vs elimination
Simultaneous equations (called 'system of equations' in US textbooks — same thing) are a set of two or more equations that must all be true at the same time. When you have two unknowns, you need two equations; three unknowns need three equations. Fewer equations than unknowns means infinitely many solutions; more equations than unknowns usually means no solution exists.
There are three standard methods, and each exam board has a favourite. Substitution — rearrange one equation to isolate a variable, then substitute that expression into the other equation. Best when one equation already has a variable with coefficient 1, like x − y = 1. Elimination — add or subtract the equations (after multiplying to match coefficients) so one variable cancels out. Best when the coefficients are already similar or easy to match. Matrix method (A⁻¹b) — write the system as a matrix equation and invert; only practical for 3×3 and up, covered in Class 12 and A-Level.
Geometrically, each linear equation in two unknowns is a straight line in the xy-plane. The simultaneous solution is the point where the lines cross. Two lines can intersect at exactly one point (unique solution), be parallel (no solution — inconsistent system), or be the same line (infinitely many solutions — dependent system). The solver flags which case you're in when there's no unique answer.
For UK students, simultaneous equations are one of the first places GCSE Maths starts asking for multiple methods on the same problem — examiners award marks for the right method even when the arithmetic slips. The solver shows both substitution and elimination side by side for that reason, so you can see which one is quicker for your problem. Three-variable systems (common in A-Level Further Maths and Class 12) use elimination to reduce to a 2-variable system, then solve as usual.
Simultaneous equations to practise (GCSE, A-Level, JEE)
Tap any problem to solve it with full step-by-step working.
- Solve with AI →1.2 variablesGCSE Year 9Easy
- Solve with AI →2.EliminationGCSE Year 10Medium
- Solve with AI →3.Linear + quadraticGCSE Year 11Hard
- Solve with AI →4.3 variablesA-Level / Class 12Hard
- Solve with AI →5.Elimination (scaling)GCSE Year 10Medium
Frequently asked questions
What's the difference between simultaneous equations and a system of equations?+
Nothing — they're the same topic. 'Simultaneous equations' is the UK/Australia name (used in AQA, Edexcel, OCR, Cambridge IGCSE syllabuses), while 'system of equations' is the US name (used in SAT, AP, Common Core). Both mean two or more equations that must all hold at once.
How do I solve simultaneous equations step by step?+
Pick a method: substitution (rearrange one equation to isolate a variable, sub into the other) or elimination (scale the two equations so a variable's coefficients match, then add or subtract to cancel it). Solve for the remaining variable, then back-substitute to find the first. Verify by plugging both values into the original equations.
When should I use substitution vs elimination?+
Use substitution when one equation already has a variable with coefficient 1 (like x − y = 5). Use elimination when both equations have similar-sized coefficients, so scaling is quick. For bigger systems (3+ variables), elimination scales better; pure substitution gets messy.
What does 'no solution' mean for simultaneous equations?+
Graphically, two linear equations with no solution represent parallel lines — they never cross. Algebraically, when you eliminate a variable you end up with a false statement like 0 = 5. The solver flags this as 'inconsistent system' and shows why.
What does 'infinitely many solutions' mean?+
The two equations represent the same line — any point on the line satisfies both. Algebraically, elimination collapses to a true statement like 0 = 0, which gives you no new information. The solver flags this as 'dependent system' and describes the solution set using a free parameter.
Can the solver handle 3 equations in 3 unknowns?+
Yes — three-variable systems are supported, shown via elimination reducing to a 2-variable system and then substituting back. This is the standard A-Level Further Maths and Class 12 approach.
Does this work for Cambridge IGCSE 0580 and 0607?+
Yes. Both syllabuses cover simultaneous linear equations (2 variables), and 0607 extends to linear + quadratic and 3-variable systems. The solver handles all three cases with the working each syllabus expects.
What if one equation is linear and the other is quadratic?+
Substitution is the standard method: solve the linear equation for one variable, substitute the expression into the quadratic, solve the resulting quadratic in one variable, then back-substitute. The solver walks through this explicitly — it's a common GCSE higher-tier and A-Level topic.
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Next
Related topics to master next
These pair naturally with simultaneous equations — each extends the same idea.
System of Equations Solver
US name for the same topic — with SAT and AP-style examples.
Open solver →
Linear Equations Solver
Master single linear equations before tackling systems.
Open solver →
Matrix Solver
The matrix method for 3+ variable systems — inversion and determinants.
Open solver →
Algebra Solver
Equation rearrangement — the skill underneath every simultaneous-equation solution.
Open solver →