Free System of Equations Solver
Paste two or three equations and see full working — substitution, elimination, or matrix inverse — all three methods walked through in plain English.
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Learn
A system solved by elimination — step by step
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
Pick a method — elimination works cleanly here
Multiply the second equation by 3 so the y-coefficients cancel when added.
- 2
Add the modified equation to the first
The y-terms cancel out, leaving a simple one-variable equation.
- 3
Solve for x
- 4
Back-substitute into either original equation
Use whichever original equation has the simpler coefficients.
- 5
Verify both equations hold
2(3) + 3(2) = 6 + 6 = 12 ✓ and 3 − 2 = 1 ✓. Both pass, so the solution is correct.
Final Answer
How to solve a system of equations — substitution, elimination and matrix methods
A system of equations is a set of two or more equations that share the same variables. The solution is the value (or values) that make every equation in the system true at the same time. For two linear equations in two unknowns, three methods work: substitution isolates one variable in one equation and plugs it into the other; elimination adds or subtracts multiples of the equations to cancel a variable; the graphical method reads off the intersection point of the two lines. For 3×3 systems, elimination scales up (row-reduce the coefficient matrix) or you can use Cramer's rule with determinants. Systems show up in SAT Heart of Algebra, GCSE simultaneous equations, CBSE Class 10 Chapter 3, and every word problem involving two unknowns — ages, mixtures, distances, prices.
Systems of equations to practise
Tap any problem to solve it with full step-by-step working.
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1. x + y = 10 and x - y = 4
Elimination (easy)Class 9 / Grade 8Easy - Solve with AI →
2. 3x + 2y = 16 and 2x - y = 4
SubstitutionClass 10 / GCSEMedium - Solve with AI →
3. x^2 + y^2 = 25 and x + y = 7
Linear + quadraticGCSE HigherMedium - Solve with AI →
4. x + y + z = 6, 2x - y + z = 3, x + 2y - z = 5
3×3 systemAlgebra IIHard - Solve with AI →
5. A ticket costs ₹x for adults and ₹y for children. 2 adults + 3 children pay ₹190; 3 adults + 2 children pay ₹210. Find x and y.
Word problemClass 10Medium
Frequently asked questions
Substitution or elimination — which is better?+
Substitution is cleaner when one equation already has a variable isolated (e.g. y = …). Elimination is cleaner when the coefficients are close to opposites or can be scaled to cancel. For 3×3 systems or messy fractions, elimination (row reduction) is almost always faster.
What does 'no solution' or 'infinitely many solutions' mean?+
No solution means the lines are parallel (same slope, different intercepts) — the system is inconsistent. Infinitely many solutions means the two equations are really the same line — the system is dependent. SolveGini flags both cases explicitly instead of silently returning wrong numbers.
Can this solve non-linear systems like x² + y² = 25 with x + y = 7?+
Yes. Substitute from the linear equation into the quadratic, reduce to a single-variable quadratic, solve, and back-substitute. The solver shows each step, including which substitution it chose and why.
Is there a matrix method and when should I use it?+
For 3×3 and larger systems, rewrite as AX = B and use the matrix inverse or Gaussian elimination. It's the go-to for A-Level Further Maths, Algebra II and engineering-style problems with many unknowns. The full row-reduction is shown when matrix form is appropriate.
Is the system of equations solver free to use?+
SolveGini has a free plan. Guests get 1 solve per day with no signup; a free account adds 5 daily solves plus quizzes, flashcards and a study planner. Step-by-step working is never paywalled.
Does it cover SAT and GCSE simultaneous equations?+
Yes. SAT Heart of Algebra, GCSE Foundation and Higher simultaneous problems, and CBSE Class 10 pair-of-linear-equations chapter — all handled with the method each curriculum prefers.
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