Solve systems of 2 or 3 linear equations with this System of Equations Calculator. Enter the coefficients for each variable (x, y, and optionally z) plus the constant on the right-hand side of each equation. The calculator returns the values of x, y, and z using Cramer's Rule, and tells you whether the system has a unique solution, no solution, or infinite solutions.
x =
--
y =
--
z =
--
Solution Type
--
Determinant (D)
--
A system of equations is a set of two or more equations that share the same variables. The goal is to find values for those variables that satisfy all equations simultaneously. For example, 2x + 3y = 8 and x − y = 1 is a 2×2 linear system with the unique solution x = 2.2, y = 1.2.
The calculator uses Cramer's Rule, a matrix-based method. It computes the determinant of the coefficient matrix and then substitutes the constants column for each variable's column to find individual determinants. Each variable equals its determinant divided by the main determinant. For a 2×2 system, this requires computing two 2×2 determinants; for 3×3, it requires four 3×3 determinants.
A linear system can have exactly one unique solution (the lines or planes intersect at a single point), no solution (the lines are parallel or the planes don't share a common point — called an inconsistent system), or infinitely many solutions (the equations describe the same line or plane — called a dependent system). The determinant of the coefficient matrix tells you which case applies: a non-zero determinant guarantees a unique solution.
If the determinant of the coefficient matrix equals zero, Cramer's Rule cannot produce a unique solution. The system is either inconsistent (no solution) or dependent (infinite solutions). You would need to inspect the augmented matrix further to determine which case applies. This calculator will flag this condition when it occurs.
A 2×2 system has two equations and two unknowns (x and y), and geometrically represents two lines in a 2D plane. A 3×3 system has three equations and three unknowns (x, y, and z), representing three planes in 3D space. Select the appropriate system size at the top of the calculator and fill in the z coefficients for the third variable when using 3×3 mode.
No — this calculator is designed specifically for linear systems, meaning all equations must be of the first degree (no x², xy, sin(x), or other non-linear terms). For non-linear systems, tools like Wolfram Alpha or Symbolab offer more advanced solving capabilities.
Cramer's Rule is a theorem in linear algebra that expresses the solution of a system of linear equations with as many equations as unknowns using determinants. It's well-suited for 2×2 and 3×3 systems because it produces exact, closed-form solutions. Each variable is solved independently, making the logic transparent and easy to verify step by step.
Simply type a negative number into the coefficient field, for example enter -3 for a coefficient of −3. The calculator handles negative values correctly in all determinant computations. Make sure to enter 0 for any variable that does not appear in a particular equation rather than leaving the field blank.