How does this solve simultaneous equations?

It uses Cramer's rule, which solves a linear system using determinants. Each unknown equals the determinant of the coefficient matrix with one column replaced by the constants, divided by the determinant of the full coefficient matrix.

What if there is no unique solution?

If the main determinant is zero, the equations either have no solution or infinitely many, so there is no single answer. The calculator reports this rather than dividing by zero.

What format do I enter equations in?

Enter the coefficients of x, y and z and the constant on the right of each equation. For 2x plus 3y equals 8, you enter 2, 3 and 8.

System of Equations Solver

This solver finds the values of the unknowns that satisfy a set of simultaneous linear equations, for either two equations in x and y, or three equations in x, y and z. Simultaneous equations turn up whenever two or more conditions must hold at the same time, such as mixing problems, working out two prices from two combined purchases, balancing supply and demand, or finding where lines and planes cross. You choose whether you have a two variable or three variable system, then enter the coefficient of each unknown and the constant on the right hand side of every equation. The calculator solves the system using Cramer's rule, a clean method that expresses each unknown as a ratio of determinants, and shows the value of the main determinant so you can see how the answer was reached. If that main determinant is zero, the equations do not have a single unique solution, meaning they are either inconsistent or describe the same relationship more than once, and the calculator tells you this instead of returning a misleading number. The method gives exact results for well behaved systems and updates the moment you change a coefficient. Enter your equations to see the solution for x, y and any z instantly.

System size

x + y + z =

x = 1, y = 2

determinant = -5

How it works

The calculator builds the coefficient matrix and works out its determinant. For each unknown it replaces the matching column with the constants, takes that determinant, and divides by the main determinant. This is Cramer's rule. When the main determinant is zero, division is not possible and the system has no unique solution.

Worked example

For 2x plus 3y equals 8 and x minus y equals minus 1, the main determinant is 2 times minus 1 minus 1 times 3, which is minus 5. Solving gives x equals 1 and y equals 2, which satisfies both equations.

Related calculators

Quadratic Equation Calculator: quadratics.
Slope Intercept Calculator: equation of a line.
Percentage Calculator: percentages.
Scientific Calculator: general maths.

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