Simultaneous equations calculator

Two simultaneous linear equations in two unknowns are a pair of equations that must both hold at the same time. We look for values of x and y that make both of them true. The calculator solves them with Cramer’s rule: it works out three determinants and divides them.

Write the equations in the form ax + by = c, with the unknowns on the left and the number on the right. If y is missing from an equation, enter a zero for its coefficient. For the equation x − y = −1, you enter a₂ = 1, b₂ = −1 and c₂ = −1.

How it works

By Cramer’s rule,

$$x=\frac{D_x}{D},y=\frac{D_y}{D}$$

where

$$D=a_1{b}_2-a_2{b}_1,D_x=c_1{b}_2-c_2{b}_1,D_y=a_1{c}_2-a_2{c}_1$$

For the system we started with,

$$\begin{array}{rcl}2x+3y& =& 8\\ x-y& =& -1\end{array}$$

we work out the determinants. The coefficients of the unknowns go into D, and for D_x and D_y we replace the column of x, or y, respectively, with the right-hand sides:

$$D=2\cdot (-1)-1\cdot 3=-5$$

$$D_x=8\cdot (-1)-(-1)\cdot 3=-5$$

$$D_y=2\cdot (-1)-1\cdot 8=-10$$

And we divide:

$$x=\frac{-5}{-5}=1,y=\frac{-10}{-5}=2$$

Check: 2 · 1 + 3 · 2 = 8 and 1 − 2 = −1. Both are true.

If D = 0, we cannot divide, and the system has either infinitely many solutions or none. Geometrically, these are two lines that coincide, or two parallel lines. By hand, simultaneous equations are usually solved by substitution or elimination; both methods are explained in the article on simultaneous equations.