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System of equations calculator

Enter the numbers in front of each letter and the answer to each equation. You get x, y (and z) with the working and a check.

By the Calcist team · Checked against known answers · Updated October 26, 2026

x+ y+ z=
x+ y+ z=
x+ y+ z=

Use 0 when a letter doesn't appear, and a negative number for minus: x − y means 1 and −1.

Work it out yourself first, then check.

Private: calculated on your device. Nothing you type is sent anywhere.

How the solution is found

Calcist uses Cramer's rule, which works for any system with one solution. First it finds the determinant D of the numbers in front of the letters. Then, for each letter, it swaps that letter's column for the answers and finds a new determinant. Each letter equals its new determinant divided by D.

For two equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂:

D = a₁b₂ − a₂b₁    x = (c₁b₂ − c₂b₁) ÷ D    y = (a₁c₂ − a₂c₁) ÷ D

If D is zero there's no single solution: the lines are parallel (no solution) or the same line (infinitely many).

Solving by elimination, by hand

Take 2x + 3y = 13 and x − y = 4.

  1. Multiply the second equation by 3: 3x − 3y = 12.
  2. Add it to the first: 5x = 25, so x = 5.
  3. Put x = 5 into x − y = 4: y = 1.
  4. Check in the first equation: 2(5) + 3(1) = 13 ✓

Substitution

Solve one equation for one letter (from x − y = 4, x = y + 4), put that into the other equation (2(y + 4) + 3y = 13, so 5y = 5 and y = 1), then work back to the other letter (x = 5). Both methods give the same answer as this calculator.

How we checked this

Before every update, this calculator is run in a real browser with these known answers, and it must match all of them:

  • 2x + 3y = 13 and x − y = 4 give x = 5, y = 1 ✓
  • Parallel lines have no solution ✓

Last checked October 26, 2026. Spotted something wrong? The math is done in your browser with the formula shown on this page, so you can check any result by hand.

Questions people ask

What is a system of equations?

Two or more equations that share the same unknowns. The solution is the set of values that makes all of them true at once.

Can a system have no solution?

Yes. 2x + y = 3 and 2x + y = 5 can never both be true, so there's no solution. On a graph the lines are parallel.

How do I enter x − y = 4?

Put 1 in front of x, −1 in front of y, and 4 as the answer.

Can it solve 3 equations with 3 unknowns?

Yes. Choose 3 equations and a z column appears.

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