System of equations calculator
Type your equations, get x, y and z.
How do I solve a system of two equations?
By substitution or elimination. For 2x + y = 5 and x − y = 1, adding the equations gives 3x = 6, so x = 2 and y = 1.
How does the calculator solve it?
By Gaussian elimination — the same row operations as elimination by hand, done in order — which works for 2 and 3 unknowns alike.
What if there is no solution?
Parallel lines never meet: x + y = 2 and 2x + 2y = 5 have no solution. The calculator says so.
What if there are infinitely many solutions?
When one equation is a multiple of another — x + y = 2 and 2x + 2y = 4 — every point on the line works.
How do I type the equations?
Just as you would write them, one per line: 2x + y = 5. Use x, y and z; numbers can be on either side.
How can I tell a contradiction from an identity?
Eliminate a variable and look at what is left (OpenStax Intermediate Algebra 2e, section 4.4). A false statement such as 0 = 1 means no solution; a true one such as 0 = 0 means infinitely many. In x + y = 2 and 2x + 2y = 5, doubling the first equation and subtracting leaves 0 = 1 (calculator: no solution); with 2x + 2y = 4 it leaves 0 = 0 (infinitely many).
What do three equations in three unknowns mean geometrically?
Each equation is a plane, and a solution is a point (x, y, z) on all three. The planes can meet in one point, have no common point, or share a line or a whole plane (OpenStax Intermediate Algebra 2e, section 4.4).
Why does it say it could not read my equation?
It reads numbers, the letters x, y and z, plus and minus signs and one equals sign, with terms on either side. Fractions such as x/2 and brackets such as 2(x + 1) are not read (both are rejected); write 0.5x and 2x + 2. Two equations may use only x and y.
How the solution is found
The calculator reads each equation into coefficients and a right-hand side, so 3x = 2y + 6 becomes 3x − 2y = 6, and applies Gaussian elimination to the augmented matrix (OpenStax Algebra and Trigonometry 2e, section 11.6). The row operations are swapping two rows, multiplying a row by a nonzero number, and adding a multiple of one row to another. They leave a triangle in which the last row gives the last unknown, which is then substituted upward. A row that collapses to 0 = 0 means infinitely many solutions; one that collapses to 0 = nonzero means none.
Worked example: three equations
OpenStax Example 4.32 is x − 2y + z = 3, 2x + y + z = 4 and 3x + 4y + 3z = −1. Typed in one per line, it returns x = 4, y = −1, z = −3. Substituting checks it: 4 − 2(−1) + (−3) = 3, 2(4) + (−1) + (−3) = 4 and 3(4) + 4(−1) + 3(−3) = −1. The two-unknown system 2x + y = 7, x − 2y = 6 from section 4.1 gives x = 4, y = −1.
Common mistakes
Sign slips when moving terms across the equals sign are the most common, which is why the result table substitutes the answer back into every equation you typed. Two equations cannot pin down three unknowns, so a third is needed for z. And “no solution” is not always a typing error: parallel lines or planes really have none.
Sources: OpenStax, Intermediate Algebra 2e, sections 4.1 and 4.4 (types of systems, elimination, Example 4.32); OpenStax, Algebra and Trigonometry 2e, section 11.6 “Solving Systems with Gaussian Elimination”. Related tools: matrix calculator, line equation calculator and equation solver. How we check our calculators.