Simultaneous Equations Solver With Steps
Solve two equations in x and y or three equations in x, y and z. The calculator shows Gaussian-elimination row operations, identifies unique, inconsistent and dependent systems, and checks unique answers in the original equations.
Written and checked by TheToolNet Editorial Team · Last reviewed: August 21, 2026 · How we check math tools
A system has one solution when rank(A) = rank([A|b]) = the number of unknowns. Unequal ranks mean no solution; a smaller shared rank means infinitely many.
Solve a Linear System
Choose two or three unknowns, then enter each equation in standard form. Integers and decimals are accepted.
What are simultaneous equations?
Simultaneous equations are equations that must be true at the same time for the same variable values. For two linear equations in x and y, each equation is a straight line; a unique solution is their intersection. Three linear equations in x, y and z describe planes, and a unique solution is the point shared by all three.
This solver accepts square linear systems in standard form, such as 2x + 3y = 13. It does not solve nonlinear expressions such as x² + y = 5, trigonometric equations, inequalities, or systems with a different number of equations and unknowns.
How the elimination method works
Gaussian elimination writes the coefficients and constants as an augmented matrix [A|b]. It uses three reversible row operations: swap two rows, multiply a row by a non-zero number, or add a multiple of one row to another. These operations change the appearance of the system without changing its solutions.
- Choose a non-zero pivot in the first variable column.
- Scale the pivot row and eliminate that variable from the other rows.
- Repeat in the remaining variable columns.
- Read the solution from reduced row-echelon form, then substitute it into every original equation.
The calculator uses partial pivoting: it selects the largest available absolute coefficient in a pivot column. This reduces avoidable rounding error for decimal inputs.
Worked simultaneous-equation examples
Two equations
For 2x + 3y = 13 and x − y = −1, the second equation gives x = y − 1. Substitution into the first gives 2(y − 1) + 3y = 13, so 5y = 15, y = 3 and x = 2. Both checks pass: 2(2) + 3(3) = 13 and 2 − 3 = −1.
Three equations
For x + y + z = 6, 2x − y + z = 3, and x + 2y − z = 2, elimination gives x = 1, y = 2 and z = 3. The 3×3 preset shows every row operation and verifies all three original equations.
Unique, no-solution and infinite-solution systems
| Reduced system | Rank test | Meaning |
|---|---|---|
| A pivot for every unknown | rank(A) = rank([A|b]) = n | One unique solution. |
| A contradiction such as 0 = 1 | rank(A) < rank([A|b]) | No solution; the system is inconsistent. |
| At least one free variable and no contradiction | rank(A) = rank([A|b]) < n | Infinitely many solutions. |
A zero determinant tells you a square coefficient matrix has no unique solution, but it does not by itself distinguish no solution from infinitely many. Comparing both ranks makes that distinction.
Choosing a solving method
| Method | Best use | Important limitation |
|---|---|---|
| Elimination | General hand calculation and transparent row steps | Fractions can grow during manual work. |
| Substitution | One variable is already isolated or has coefficient 1 | Becomes cumbersome for larger systems. |
| Cramer's rule | Small square systems with non-zero determinant | Does not handle singular systems by a direct solution formula. |
| Graphing | Visualizing two-variable systems | Reading an exact intersection from a graph can be imprecise. |
Frequently asked questions
How do you solve simultaneous equations?
Eliminate one variable by adding or subtracting suitable multiples of equations. Continue until the remaining variables can be read or back-substituted, then check the answer in every original equation.
Does this solver handle three simultaneous equations?
Yes. Select the 3×3 option for three linear equations in x, y and z. The calculator reduces the augmented matrix and verifies unique solutions.
Why can simultaneous equations have no solution?
The equations may conflict. In two dimensions this often means distinct parallel lines; algebraically, elimination produces a contradiction such as 0 = 1.
Why can there be infinitely many solutions?
One or more equations may be redundant, leaving fewer independent constraints than unknowns. Elimination then leaves at least one free variable without a contradiction.
Are decimal answers exact?
No. Decimal inputs and non-integer results are displayed as numerical approximations. Use the reported substitution residual to judge the numerical check.
Related tools and software guides
For spreadsheet work, see how to solve simultaneous equations in Excel. For numerical or symbolic code, use our MATLAB simultaneous-equations guide. The matrix diagonalization calculator handles a different matrix task involving eigenvalues and eigenvectors.