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Calculator Answers

System of Equations Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A system of equations calculator is an algebraic tool that solves sets of simultaneous equations with multiple variables. The calculator supports systems of two or three linear equations, using methods such as substitution, elimination, matrix inversion, or Cramer's rule to find the variables' values. It can identify unique solutions, infinitely many solutions, or inconsistent systems with no solution, and displays the step-by-step mathematical work for the selected resolution method. Students and engineers use this tool to solve linear algebra problems.

Pick the size, enter each equation in standard form (coefficients on the left, constant on the right), and the calculator returns the solution using Cramer's rule. No-solution and infinite-solution cases are reported.

Quick Answer

Solve simultaneous linear equations. Enter your equations with two or three variables to find the unique solution with step-by-step work.

System size

Equation 1: a₁ x + b₁ y = c₁

Equation 2: a₂ x + b₂ y = c₂

Solution

Solution

x = 2, y = 1

Determinant = -7

MethodCramer's rule: x = (c₁b₂ − c₂b₁) / det = 2, y = (a₁c₂ − a₂c₁) / det = 1.

For larger systems or to see the elimination steps explicitly, switch to the RREF calculator and paste the augmented matrix.

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Examples

2x + y = 5; x − 3y = -1

x = 2, y = 1

x + y + z = 6; 2y + 5z = -4; 2x + 5y − z = 27

x = 5, y = 3, z = -2

x + 2y = 4; 2x + 4y = 9

no solution (parallel)

x + 2y = 4; 2x + 4y = 8

infinite solutions (same line)

How it works

The calculator computes the coefficient determinant D and, for each variable, a replaced-column determinant. The unique solution (when D ≠ 0) is the ratio of these determinants.

2×2 case · x = (c₁b₂ − c₂b₁) / (a₁b₂ − a₂b₁)

3×3 case · xᵢ = det(Aᵢ) / det(A)

When det(A) = 0, the system has either no solution or infinitely many; the calculator distinguishes the 2×2 case by checking row proportionality.

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More tools from Education

RREF CalculatorConvert a 2x2, 2x3, 3x3, or 3x4 matrix to reduced row echelon form using Gauss-Jordan elimination. Returns the RREF matrix, the matrix rank, and a solution reading for augmented systems.

Frequently asked questions

A set of equations involving the same variables. A solution is a set of variable values that satisfies every equation at once. In 2D, two linear equations describe two lines; their intersection is the solution.

For a system with coefficient matrix A and right-hand side b, each variable x_i equals det(A_i) / det(A), where A_i is A with the i-th column replaced by b. The method requires det(A) ≠ 0; when it is zero, the system is either inconsistent or dependent.

Two lines that are parallel never meet, so the system has no solution. The calculator detects this by checking whether the rows are proportional but with different constants on the right.

Two lines that are exactly the same have every point in common, so the system has infinitely many solutions. The calculator flags this case but does not parameterize the solution set; for that, use the RREF calculator.

Use this calculator for a fast x, y (or x, y, z) answer to a typical homework or word-problem system. Use RREF when you want to see the elimination steps, or when you suspect the system is dependent and you want to read off the free variables.

Yes. Coefficients can be any real number. Negative leading coefficients are handled correctly.