Simultaneous Equations Solver

Simultaneous equations are two or more equations that must all be true at the same time. This solver handles systems of two equations in x and y or three in x, y and z. It uses elimination with exact fractions, spots systems with no solution or infinitely many, and graphs two-variable systems.

Use x and y (and z for three equations).
Try:

How to use the simultaneous equations solver

  1. Type each equation on its own line, such as 2x + 3y = 13.
  2. Use x and y, or x, y and z for three equations.
  3. Press Calculate to see the solution, the steps and the graph.

Formula

{a1x+b1y=c1a2x+b2y=c2\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}

Worked example: 2x + 3y = 13, x − y = −1

  • Equations (one per line): 2x + 3y = 13 x - y = -1

✓ Answer checked

x=2,y=3x = 2,\quad y = 3
−4−22468−22468(2, 3)2x + 3y = 13x - y = -1
Each equation is a straight line; the solution is where they cross.
Step-by-step working (3 steps)

  1. Write the augmented matrix

    Put the coefficients and constants into a matrix.

    (23131−1−1)\begin{gathered}\left(\begin{array}{cc|c}2 & 3 & 13 \\ 1 & -1 & -1\end{array}\right)\end{gathered}

  2. Eliminate Gauss–Jordan elimination

    Use row operations to clear the variables one at a time (elimination).

    R1→12 R1R2→R2−1 R1R2→−25 R2R1→R1−32 R2\begin{gathered}R_{1} \to \frac{1}{2}\,R_{1}\\R_{2} \to R_{2} - 1\,R_{1}\\R_{2} \to -\frac{2}{5}\,R_{2}\\R_{1} \to R_{1} - \frac{3}{2}\,R_{2}\end{gathered}

  3. Read the solution

    Each row now gives one variable.

    x=2,y=3x = 2,\quad y = 3
How this was checked
  • ✓ Substituting the solution into every original equation makes both sides equal.

Frequently asked questions

How does elimination work?

Add or subtract multiples of the equations so that one variable cancels, solve for the other, then substitute back.

What if the lines are parallel?

Parallel lines never meet, so the system has no solution. The calculator detects this and shows it on the graph.

Can it solve three equations?

Yes. Enter three equations using x, y and z, one per line.

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