About the System of Equations Solver
This system of equations solver finds the values of x and y (or x, y and z) that satisfy two or three linear equations at the same time. Enter the coefficients of each equation in the form ax + by + cz = d and the solver returns the unique solution, or tells you when the system has no solution (inconsistent, parallel lines or planes) or infinitely many solutions (dependent equations).
It is built for algebra and linear-algebra students checking homework, teachers preparing worked examples, and anyone solving practical mixture, pricing, break-even or circuit problems that reduce to simultaneous equations. The step table shows the main determinant and the determinants for each variable, exactly as in Cramer’s rule, so you can follow the working and copy it into your notes.
Coefficients can be any real numbers, including decimals and negatives. For a 2×2 system only the x and y columns of the first two equations are used; switch to 3×3 to add the z column and a third equation.
How to use the system of equations solver
- 1Choose a 2×2 or 3×3 system.
- 2Rewrite each equation in the form ax + by (+ cz) = d, moving all variables to the left.
- 3Enter the coefficients and constants; use 0 for a missing variable and negative numbers for subtraction.
- 4Read the solution, and check the determinant table to see Cramer’s rule step by step.
- 5If the solver reports no or infinitely many solutions, check whether equations are multiples of each other.
Formula and method
Cramer’s rule solves a square linear system using determinants. D is the determinant of the coefficient matrix. To find a variable, replace that variable’s column of coefficients with the right-hand-side constants, take the determinant of the new matrix (Dₓ, D_y or D_z) and divide it by D. For a 2×2 system D = a₁b₂ − a₂b₁; for a 3×3 system the determinant is expanded along the first row.
If D ≠ 0 the system has exactly one solution. If D = 0 the equations are not independent: the solver then compares the rank of the coefficient matrix with the rank of the augmented matrix (Gaussian elimination). Equal ranks mean infinitely many solutions (dependent equations); a higher augmented rank means the equations contradict each other and there is no solution.
- aᵢ, bᵢ, cᵢ
- Coefficients of x, y, z in equation i
- dᵢ
- Constant on the right-hand side of equation i
- D
- Determinant of the coefficient matrix
- Dₓ, D_y, D_z
- Determinants with one column replaced by the constants
Worked examples
2×2: 2x + 3y = 8 and x − y = −1
D = 2·(−1) − 3·1 = −5. Dₓ = 8·(−1) − 3·(−1) = −5, so x = 1; D_y = 2·(−1) − 8·1 = −10, so y = 2. Check: 2 + 6 = 8 and 1 − 2 = −1.
3×3 classic: 2x + y − z = 8, −3x − y + 2z = −11, −2x + y + 2z = −3
The coefficient determinant is −1, so there is one solution. Cramer’s rule gives x = 2, y = 3 and z = −1, which satisfy all three equations (e.g. 4 + 3 + 1 = 8).
Elimination-style pair: 3x − 2y = 7 and 5x + 4y = 19
D = 3·4 − (−2)·5 = 22. Dₓ = 7·4 − (−2)·19 = 66, so x = 3; D_y = 3·19 − 7·5 = 22, so y = 1.
Parallel lines: x + y = 2 and 2x + 2y = 5
The second equation’s left side is exactly double the first, but 5 is not double 2. The lines are parallel, D = 0, and no pair (x, y) satisfies both.
Same line twice: x + y = 2 and 2x + 2y = 4
The second equation is just the first multiplied by 2, so both describe the same line and every point on it (such as x = 0, y = 2 or x = 1, y = 1) is a solution.
Frequently asked questions
What are the methods for solving a system of equations?+
The main methods are substitution, elimination (adding or subtracting equations), graphing, matrices with Gaussian elimination, and Cramer’s rule with determinants. They all give the same answer; this solver uses Cramer’s rule and falls back to elimination to classify singular systems.
How do I know if a system has no solution?+
A linear system has no solution when its equations contradict each other — for two unknowns, the lines are parallel with different intercepts. In that case the determinant is 0 and elimination produces a false statement such as 0 = 1.
What does infinitely many solutions mean?+
It means the equations are not independent: one can be built from the others, so they describe the same line or plane. Any point on that shared line or plane satisfies every equation.
When can I use Cramer’s rule?+
Cramer’s rule works for square systems (same number of equations as unknowns) whose coefficient determinant is not zero. It is ideal for 2×2 and 3×3 systems; for larger systems Gaussian elimination is faster.
How do I enter an equation with a missing variable?+
Use 0 as the coefficient. For example, in a 3×3 system the equation x + 2z = 5 is entered as x coefficient 1, y coefficient 0, z coefficient 2 and right-hand side 5.