Systems Of Equations Calculator

Background.

Plenty of everyday puzzles have two unknowns and two facts that tie them together. A cinema sold 30 tickets for $300; adults pay $12 and children $8. How many of each? Each fact is a linear equation, and the answer is the one pair of numbers that satisfies both at once.

Drawn on a graph, each equation is a straight line, and the solution is the point where the two lines cross. That picture explains the three ways a pair of equations can turn out. Lines with different slopes cross exactly once, giving a single answer. Parallel lines never meet, so no pair works and the equations contradict each other. Two equations that are the same line in disguise, one a multiple of the other, agree everywhere and leave infinitely many answers.

The page solves the pair with Cramer's rule, which reduces the problem to three small determinants. The key one, a₁b₂ − a₂b₁, is built from the x and y coefficients alone and decides which case you are in. If it is anything but zero, the lines cross once and the solution follows directly. If it is zero, the lines are parallel or identical, and the page returns an error instead of inventing an answer; it does not say which of the two it is.

A determinant that is not zero but tiny next to the coefficients is a warning of another kind. The lines are nearly parallel, and small changes in the constants can swing the crossing point a long way, so answers built on measured or rounded data deserve caution.

Before entering anything, put each equation in the form ax + by = c, unknowns on the left and the constant on the right. A missing variable has coefficient 0, and a term moved across the equals sign changes sign.

What is systems of equations calculator?

A system of two linear equations in two unknowns is a pair of statements, a₁x + b₁y = c₁ and a₂x + b₂y = c₂, that must hold at the same time. Solving it means finding the x and y that make both true. When the coefficient determinant a₁b₂ − a₂b₁ is non-zero there is exactly one such pair, the point where the two lines intersect; when it is zero the system has either no solution or infinitely many.

How to use this calculator.

1. 01Rewrite each equation as ax + by = c. For y = 3x − 2, subtract 3x from both sides to get −3x + y = −2, so the x coefficient is −3, the y coefficient 1 and the constant −2.
2. 02Enter the first equation's x coefficient, y coefficient and constant, then the second equation's. Use 0 for a variable that does not appear, and keep every minus sign.
3. 03Read off x and y, then substitute them back into both original equations. It takes seconds and catches sign errors made while rearranging.
4. 04Look at the determinant. If the page refuses the system, the lines are parallel or coincide; if the determinant is very small compared with the coefficients, treat the answer as sensitive to rounding in the inputs.

The formula.

With D = a₁b₂ − a₂b₁, Cramer's rule gives x = (c₁b₂ − c₂b₁) ÷ D and y = (a₁c₂ − a₂c₁) ÷ D. Each numerator is the determinant D with one column of coefficients swapped for the constants: the x column when solving for x, the y column when solving for y. D measures how far the two lines are from parallel. It is zero exactly when one row of coefficients is a multiple of the other, and then the division cannot be done, so the page stops with an error. Elimination and substitution reach the same x and y, since for two equations the methods are algebraically equivalent. For large systems Cramer's rule becomes very costly and Gaussian elimination is the usual choice, but for a 2 × 2 system it is the most direct route. Coefficients and constants can be any numbers from −1,000,000 to 1,000,000, decimals and negatives included.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.