Simultaneous Equations Solver
Solve simultaneous equations step by step by elimination or substitution. Linear systems with up to 6 unknowns and nonlinear systems with x², y² and xy, with exact answers, a check and a graph.
How Simultaneous Equations Are Solved
Multiply two equations by numbers chosen so that one unknown has opposite (or equal) coefficients, then add (or subtract) them. That unknown disappears. Repeat until one equation in one unknown is left, solve it, and substitute back. For 5x + 3y = 7 and 3x − 5y = −23, 3·(1) − 5·(2) gives 34y = 136, so y = 4, and then 3x − 20 = −23 gives x = −1.
Solve one equation for one unknown and put the result into the other equations. This is the natural method for nonlinear systems when one equation is linear: from x + y = 7, y = 7 − x, and x² + y² = 25 becomes x² + (7 − x)² = 25, a quadratic with roots x = 3 and x = 4.
A linear system has exactly one solution, no solution, or infinitely many. Two equations of degrees m and n in two unknowns have at most m·n common solutions unless they share a common factor (Bézout's theorem). So a line and a circle meet at most twice, and two conics at most four times. In the graph, each real solution is an intersection point of the curves.
Disclaimer: This calculator performs standard mathematical operations for general and educational use only. We are not a substitute for professional review — for calculations where accuracy is critical, such as financial filings, engineering work, or academic submissions, please verify the result independently.
Frequently Asked Questions
About this calculator
This simultaneous equations solver finds the values of the unknowns that make several equations true at the same time. It solves linear systems with up to 6 equations and 6 unknowns, and nonlinear systems of 2 equations in 2 unknowns with squares and products such as x², y² and xy. Type the equations one per line or separated by commas, and choose elimination or substitution. You get the exact solutions, every step, a check, and a graph of the curves with the intersection points marked.
What are simultaneous equations?
Simultaneous equations are two or more equations that share unknowns and must all be true at once. A solution is a set of values, one for each unknown, that satisfies every equation. With two unknowns, each equation is a line or a curve in the plane, and the solutions are exactly the points where all the curves meet. For example, x + y = 10 and x − y = 2 meet at (6, 4).
Elimination method
- Multiply the equations by suitable whole numbers so that one unknown has the same coefficient in two of them.
- Subtract (or add) the equations to remove that unknown.
- Repeat until one equation in one unknown is left, and solve it.
- Substitute the value back to find the other unknowns.
Example: 5x + 3y = 7 and 3x − 5y = −23. Multiply the first equation by 3 and the second by 5, then subtract: 34y = 136, so y = 4. Then 3x − 5(4) = −23 gives x = −1. The solver keeps every new equation in whole numbers and gives it a number, so you can follow which equations were combined.
Substitution method
Solve one equation for one unknown, then put that expression into the other equations. Each substitution removes one unknown. For x + 2y = 2x − 5 and x − y = 3, the second equation gives x = y + 3. Putting this into the first equation gives (y + 3) + 2y = 2(y + 3) − 5, so y = −2 and x = 1.
Nonlinear systems
When an equation contains x², y², xy or higher powers, the system is nonlinear and can have several solutions. The solver picks the simplest of three routes and tells you which one it used.
- Substitution: if one equation is linear in an unknown, such as y = 2x − 1, it is solved for that unknown and put into the other equation. y = x² − 4 and y = 2x − 1 give x² − 2x − 3 = 0, so x = 3 or x = −1, and the solutions are (3, 5) and (−1, −3).
- Elimination: if the terms with one unknown are the same in both equations, subtracting or adding removes it. For x² + y² = 13 and x² − y² = 5, adding gives 2x² = 18, so x = ±3 and y = ±2: four solutions.
- Resultant: in other cases, the resultant of the two polynomials removes one unknown and leaves one polynomial equation in the other unknown.
Exact answers are kept whenever possible. For x² + y = 5 and x² + y² = 7, the solutions are x = ±√3, y = 2 and x = ±√6, y = −1. If there are no real solutions, as with x² + y² = 4 and x + y = 5 (the line misses the circle), the solver says so.
How many solutions can there be?
A linear system has exactly one solution, no solution, or infinitely many. Two equations of degrees m and n in two unknowns have at most m × n common solutions, unless they share a common factor (Bézout's theorem). So a line and a parabola meet at most twice, and two circles or other conics meet at most four times. If one equation is a multiple of the other, or they share a factor, there are infinitely many solutions.
Where simultaneous equations are used
- Prices: 2 coffees and 3 muffins cost 13, and 4 coffees and 1 muffin cost 11. So 2c + 3m = 13 and 4c + m = 11, which gives c = 2 and m = 3.
- Geometry: a rectangle with perimeter 34 and area 60 has sides with x + y = 17 and xy = 60, so the sides are 5 and 12.
- Science and engineering: currents in electric circuits, forces in a truss and balanced chemical equations all lead to systems of linear equations.
- Intersections: the meeting points of a line and a circle, or of two curved paths, are the solutions of a nonlinear system.
- Linear and nonlinear systems — Solves linear systems with up to 6 equations and 6 unknowns, and nonlinear systems of 2 equations in 2 unknowns with terms such as x², y² and xy.
- Elimination or substitution — Choose the method. Elimination combines equations with whole-number multipliers and numbers each new equation. Substitution solves for one unknown at a time.
- Exact solutions — Answers are given as whole numbers, fractions or square roots such as √3, with decimals next to them.
- Every case covered — Shows when a system has no solution, no real solution or infinitely many solutions, and gives the general solution in terms of the free unknowns.
- Graph and check — Plots each equation as a line or curve with the intersection points marked, and puts every solution back into every equation.