\ begin smallmatrix x ; y end smallmatrix.Algebra · Simultaneous equations
System of Equations Solver by any method
Solve simultaneous equations step by step, by substitution, elimination, Cramer's rule or an inverse matrix — in exact fractions, with a graph that shows where the lines meet.
- Free steps
- Graph
- Checked answers
Try it
Tilt the lines. Find where they meet.
Each linear equation is a line; the solution is where the lines cross. Compare the gradients to predict 1, 0 or infinitely many solutions.
System
(1) y = 0.5x + 1
(2) y = -x + 4
gradients 0.5 and -1 — different
Different gradients, so the lines cross exactly once: one solution — the intersection point.
Drag a line, or focus it and press ↑ ↓, to slide it; the sliders tilt it.
Why this system of equations solver is different
01
Your method, not ours
Choose elimination, substitution, Cramer’s rule or the matrix method and get that exact working — the one your teacher or exam board expects — with fractions kept exact and every bracket expanded.
02
Every case explained
One solution, no solution or infinitely many: you see why — crossing, parallel or identical lines, planes sharing a line — with a parametric answer for free variables.
03
Checked, then graphed
The answer is substituted back into every original equation before the Verified stamp appears, and two-variable systems are drawn with the intersection marked.
What is a system of equations solver?
A system of equations solver finds the values of the unknowns — usually x, y and z — that make every equation in a set true at the same time. Such a set is called a system of equations or simultaneous equations.
With two unknowns each linear equation is a straight line, and the solution is where the lines cross. Lines can cross once (one solution), be parallel (no solution) or be the same line (infinitely many) — the gradients tell you which. With three unknowns each equation is a plane.
Simultaneous equations are core to GCSE, CBSE Class 10 and 12, A-level, AP Precalculus and the IB, so the working is shown in every standard method — pick the one your class uses.
How to use the system of equations solver
- 1Type one equation per row, for example 2x + 3y = 7 and x − y = 1. Use Add equation for a third or fourth row, or paste LaTeX from a worksheet.
- 2Choose a method: Elimination (the default), Substitution, Cramer’s rule or Matrix. Cramer’s rule and the matrix method work for square systems up to 3 × 3.
- 3Press Solve. The answer card gives each unknown as an exact fraction (with a decimal when useful) and says whether the system has one, none or infinitely many solutions.
- 4Open the steps to follow the working, the Graph tab to see the lines and their intersection, and the Table tab to see the answer substituted into each equation.
- 5Tilt the lines in the interactive graph, then press Solve this system with steps to send them to the solver.
Elimination, substitution, Cramer’s rule and matrices
All four methods give the same answer; they differ in how much arithmetic they need.
Elimination method
Multiply the equations so one variable has equal (or opposite) coefficients, then subtract (or add) to eliminate it. For three equations the solver does the same on the augmented matrix, labelling every row operation such as R_2 → R_2 - 2R_1.
Substitution method
Make one variable the subject of one equation and substitute that expression, in brackets, into the others. It is quickest when a variable already has coefficient 1.
Cramer’s rule
D is the determinant of the coefficients; D_x replaces the x-column with the constants. If D = 0 the rule cannot be used.
Matrix method (inverse matrix)
The CBSE Class 12 method: find |A|, the cofactors and adj A, then multiply. If |A| = 0 and (adj A)B ≠ O, the system is inconsistent.
How many solutions? Compare the gradients
Different gradients mean the lines cross once. Equal gradients with different intercepts mean parallel lines; if all three ratios are equal, both equations describe the same line.
Nonlinear systems: a circle and a line
Solve the linear equation for one variable and substitute into the other to get a quadratic. Its discriminant tells you whether the line cuts, touches or misses the circle.
System of equations solver worked example: 2x + 3y = 7 and x − y = 1
- 01
The x-coefficients are 2 and 1, so multiply equation (2) by 2 to match them.
2x - 2y = 2 - 02
The x-terms are now equal, so subtract this from equation (1) to eliminate x.
(2x + 3y) - (2x - 2y) = 7 - 2 ⇒ 5y = 5 - 03
Divide both sides by 5.
y = 1 - 04
Substitute y = 1 into equation (2) and solve for x.
x - 1 = 1 ⇒ x = 2 - 05
Check in both original equations: 2(2) + 3(1) = 7 and 2 - 1 = 1. ✓ The gradients -2/3 and 1 differ, so this is the only solution.
Common mistakes with simultaneous equations
Multiplying only one side
When you scale an equation, multiply every term on both sides: 2 × (x - y = 1) is 2x - 2y = 2, not 2x - y = 1.
Sign errors when subtracting
Subtracting (2x - 2y) from (2x + 3y) gives 5y, not y: the minus sign applies to the whole bracket. If the coefficients are opposites, add instead.
Stopping after one variable
Finding y is only half the job — substitute it back to get x, and give the answer as an ordered pair (x, y).
Using Cramer’s rule when D = 0
A zero determinant means the lines are parallel or identical, so D_x/D is undefined. Decide between “no solution” and “infinitely many” by elimination.
System of equations solver FAQ
Is this solver free to use?+
Yes. The step-by-step engine on this page — every method, the graph, the substitution check and the practice problems — is free to use without signing up, and your equations are solved right in your browser. Only the optional AI tutor, which answers follow-up questions in conversation, has a daily allowance.
Can it solve 3 equations with 3 unknowns?+
Yes. Add a third equation with Add equation. The solver uses Gaussian elimination with labelled row operations (clearing any fractions first), substitution, Cramer’s rule with 3 × 3 determinants, or the matrix method with cofactors and the adjoint. Systems with four unknowns are solved by elimination.
How do I know if a system has no solution or infinitely many?+
If elimination leads to a false statement such as 0 = 5, there is no solution: the lines are parallel, with the same gradient but different intercepts. If it leads to 0 = 0, the equations are dependent and there are infinitely many solutions, written in parametric form, for example (x, y) = (2 - t, t).
Which method should I use for my exam?+
Use the method your syllabus asks for. GCSE and CBSE Class 10 questions usually expect elimination or substitution; CBSE Class 12 asks for the matrix method X = A^(-1)B; Cramer’s rule appears in many algebra and linear-algebra courses. Switch the Method option to see each one on the same system.
Does it work with fractions and decimals?+
Yes. Coefficients such as x/2 or 0.25y are converted to exact fractions, and the first step multiplies each equation through to clear them. Answers stay exact, for example x = 17/11, with a decimal shown alongside so you can compare it with your calculator.
Can it solve nonlinear systems like a circle and a line?+
Yes, for two equations in two unknowns. It names each curve (for example a circle with centre (0, 0) and radius 5), substitutes the linear equation into the other and solves the quadratic exactly, surds included. When the line misses the circle, the distance from the centre proves it.
How is the answer verified?+
The solution is substituted into every original equation and both sides are compared. For “no solution”, the solver checks a combination of the original equations that reduces to a contradiction such as 0 = 1, so the stamp never relies on the method that produced the answer.
Can I paste equations from a worksheet or LaTeX?+
Yes. Paste equations separated by semicolons, commas or new lines. LaTeX such as \frac{x}{2} + y = 4 or a \begin{cases} block is understood, as are signs like − and ×. An augmented matrix such as [[2, 3, 7], [1, -1, 1]] is read as 2x + 3y = 7, x - y = 1.
Stuck on a different problem? Snap it.
Photo, typed or pasted — the maths solver reads it, solves it and checks the answer.