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System of Equations Calculator

Solve systems of two or three linear equations. Get exact fraction answers, Cramer’s rule worked step by step, a check of every equation and a graph of the lines.

100% Private Exact fractions Graph for 2 × 2

Your Equations

Use whole numbers, decimals (0.5) or fractions (1/2). Leave a box empty for 0.
Solution
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Step by step (Cramer’s rule)

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Check

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Graph

Reduced row echelon form

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How to solve a system of equations

A system of linear equations is a set of equations that share the same unknowns. Its solution is the set of values that makes every equation true at once. There are three common methods:

  • Substitution: solve one equation for a variable and substitute it into the others.
  • Elimination: add or subtract multiples of the equations to cancel a variable.
  • Cramer’s rule: use determinants — the method shown step by step above.

Cramer’s rule for two equations

a₁x + b₁y = c₁
a₂x + b₂y = c₂

D  = a₁b₂ − b₁a₂
Dₓ = c₁b₂ − b₁c₂  ⇒ x = Dₓ ÷ D
Dᵣ = a₁c₂ − c₁a₂  ⇒ y = Dᵣ ÷ D

If D = 0 the equations do not have a single solution: the lines are either parallel (no solution) or the same line (infinitely many solutions). The reduced row echelon form tells which.

Frequently Asked Questions

Exactly one, none, or infinitely many. With two equations in two unknowns, these are lines that cross at one point, parallel lines, or the same line drawn twice.

A zero determinant means the equations are not independent, so there is no single unique solution. The system is then either inconsistent (no solution) or dependent (infinitely many solutions).

Yes. Type 3/4, -2/5 or 0.75. The calculator works with exact fractions internally, so answers like x = 7/3 are shown exactly, with a decimal next to them.

Use 0 for its coefficient, or leave the box empty. For example, 2x + 5 = 9 with no y term is entered as 2, 0 and 9.

Rearrange them first so all the variables are on the left and the number is on the right. For example, y = 3x − 2 becomes −3x + y = −2.
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