2x + y = 7, x − y = 2
- 1.Add the equations — the y terms cancel: 3x = 9
- 2.x = 3
- 3.Back-substitute into x − y = 2: 3 − y = 2, so y = 1
x = 3, y = 1
Verified in both: 2(3) + 1 = 7 ✓ and 3 − 1 = 2 ✓.
System of equations, solved — and checked against both equations
Type two equations separated by a comma. You get the exact solution, the elimination steps, and a proof: the answer substituted back into both originals. That double check is what makes a system answer trustworthy.
Scope: systems of two linear equations in x and y. Three-equation systems and non-linear systems are not covered yet — the solver will say so rather than guess.
- ✓100% free — no paywall, ever
- ✓No sign-up, no app
- ✓Every answer verified before you see it
In short
A system of equations calculator finds the single pair (x, y) that makes two equations true at the same time. VerifySolve solves two linear equations by elimination with exact fractions, shows every step, and proves the solution by substituting it back into both original equations — a real solution must satisfy each one, not just one of them.
The formulas behind the steps
Elimination is just organised arithmetic: make one variable cancel, solve, back-substitute. Underneath, the solver uses the determinant form of the same idea — which is also how it spots no-solution and same-line systems instead of guessing.
The standard form
The determinant
Solve for x
Solve for y
Worked examples — solved and double-checked
Every answer below was produced by the solver above and then verified in both equations. Try them — or tap any example chip in the solver to run it yourself.
3x + 2y = 12, x + y = 5
- 1.Double the second equation: 2x + 2y = 10
- 2.Subtract from the first: x = 2
- 3.Back-substitute: 2 + y = 5, so y = 3
x = 2, y = 3
Verified in both: 3(2) + 2(3) = 12 ✓ and 2 + 3 = 5 ✓.
4x + 3y = 18, 2x − y = 2
- 1.Triple the second equation: 6x − 3y = 6
- 2.Add to the first: 10x = 24, so x = 12/5
- 3.Back-substitute: y = 14/5
x = 12/5, y = 14/5
Real systems often land on fractions — exact answers, no rounding. Verified: 4(12/5) + 3(14/5) = 18 ✓ and 2(12/5) − 14/5 = 2 ✓.
Word form: two numbers add to 20 and differ by 4
- 1.Translate: x + y = 20 and x − y = 4
- 2.Add the equations: 2x = 24, so x = 12
- 3.Then y = 20 − 12 = 8
the numbers are 12 and 8
Verified against the story: 12 + 8 = 20 ✓ and 12 − 8 = 4 ✓.
Common mistakes — and how to dodge them
Solving for x… and stopping
A system is not solved until you have both values. The most common lost mark in class: finding x = 3, boxing it, and never back-substituting for y. The answer is a pair.
Sign slips while eliminating
When you subtract one equation from another, every term flips sign — including the constant. Subtracting x − y = 2 means −x + y = −2. Our steps write the scaled equation out in full so the signs stay visible.
Assuming every system has one answer
Parallel lines never meet (no solution) and identical lines meet everywhere (infinitely many). If your elimination collapses to 0 = 5 or 0 = 0, that IS the answer — not a mistake.
System of Equations Calculator — questions
How do you solve a system of two equations?
The most reliable hand method is elimination: scale one or both equations so a variable cancels when you add them, solve for the remaining variable, then substitute back. Substitution works well when a variable is already isolated. This calculator uses elimination with exact fractions — and shows every step.
What does the solution of a system actually mean?
It is the one pair of values (x, y) that makes both equations true at the same time. Graphically it is the point where the two lines cross. That is why our verification plugs the answer into both original equations — a real solution has to satisfy each one.
What if my system has no solution, or infinitely many?
That happens when the two equations describe parallel lines (no solution) or the exact same line (infinitely many). The solver detects this and tells you plainly instead of inventing an answer — some calculators silently return nonsense here.
Are the answers exact, or rounded decimals?
Exact. VerifySolve computes with fractions internally, so an answer like x = 12/5 stays 12/5 — never 2.3999999. Decimals in your input are converted to exact fractions too.
Can it solve systems of three equations?
Not yet — right now the solver covers two equations in two unknowns (x and y), which is the standard Algebra 1 and 2 case. Three-equation systems are on the roadmap.
Is the system of equations calculator free?
Yes — completely free, with no sign-up, no app download and no step paywall. Type the two equations separated by a comma, for example "2x + y = 7, x - y = 2".
Keep solving
Equation Solver →
Any single equation — linear or quadratic — routed to the right method automatically.
Algebra Calculator →
The all-rounder: linear equations, quadratics, systems, simplifying and factoring.
Word Problems →
Story problems in stages — including the “two numbers” problems that become systems.
Solve for X Calculator →
Focused practice on isolating x in one equation.
A note on honest use: VerifySolve is built for learning. Use the steps to understand the method, then close the tab and try one on your own — that is where the grade actually comes from. If your teacher or school has a policy on calculators, follow it.