Simultaneous Equations Solver
Choose two or three unknowns and enter the numbers for each equation. You get the exact solution, or a clear message if there is none or there are infinitely many.
Enter each equation as numbers in front of x, y and z, then the number on the right of the equals sign. Use 0 for a missing term.
Worked examples
Two unknowns
2x + y = 5 and x - y = 1
x = 2, y = 1
Adding the equations gives 3x = 6.
Three unknowns
x + y + z = 6, 2y + 5z = -4, 2x + 5y - z = 27
x = 5, y = 3, z = -2
Each equation checks: 5 + 3 - 2 = 6.
About Simultaneous Equations Solver
Simultaneous equations are a set of equations that must all be true at once. With two unknowns you need two equations, and with three unknowns you need three. They appear whenever two quantities depend on each other: the prices of two items from two different receipts, the amounts in a mixture, or the speeds in a pair of journeys. The usual school methods are elimination and substitution, and the same ideas work for larger sets.
Choose two or three unknowns and type the numbers in front of x, y and z, and the number on the right of each equals sign. The solver uses elimination with exact fractions, so the answers are exact, and then substitutes them back into every equation so you can check each line. For two unknowns it also shows the determinants used in Cramer's rule. It recognises the two cases that have no single answer: equations that contradict each other, like parallel lines, and equations that repeat the same information, which leave infinitely many answers.
How to use Simultaneous Equations Solver
- Choose two or three unknowns.
- Write each equation with the unknowns in the same order, and enter the numbers in front of x, y and z.
- Enter the number on the right of the equals sign for each equation. Use 0 for any missing unknown.
- Read the solution, then use the check lines to confirm each equation holds.
Common uses
- Solving pairs of equations in algebra homework
- Finding prices from two shopping totals
- Balancing mixtures and quantities
- Finding where two straight lines cross
Frequently asked questions
How do I solve simultaneous equations by hand?
Multiply one or both equations so that one unknown has the same number in each, then add or subtract to remove it. Solve for the remaining unknown and substitute back. This page does the same with exact fractions.
What does no solution mean?
The equations contradict each other. For two unknowns, the lines are parallel and never meet. For example x + y = 2 and x + y = 5 cannot both be true.
What does infinitely many solutions mean?
One equation carries no new information, as with x + y = 2 and 2x + 2y = 4. They describe the same line, so every point on it works.
What is Cramer's rule?
It solves a system using determinants. For two equations, x is Dx divided by D and y is Dy divided by D, where D is the determinant of the coefficients. It only works when D is not zero.
Can the equations have decimals or fractions?
Yes. Type decimals or fractions such as 1/2 and the answers will stay exact.