About the Linear Equation System Solver
What is the Linear Equation System Solver?
A system of linear equations is a set of equations that must all be true at the same time, such as 2x + y − z = 8, −3x − y + 2z = −11 and −2x + y + 2z = −3. Each equation is a line (in 2D) or a plane (in 3D), and the solution is the point where they all meet.
Enter the coefficients of x, y and z and the right-hand side of each equation. The solver uses Cramer's rule: if the determinant of the coefficient matrix is not zero there is exactly one solution, and each unknown is a ratio of two determinants.
How to use the Linear Equation System Solver
- Number of Unknowns — choose 2 to solve for x and y using only equations 1 and 2. (sample: 3 (x, y, z))
- Eq. 1: coefficient of x (sample: 2)
- Eq. 1: coefficient of y (sample: 1)
- Eq. 1: coefficient of z (sample: -1)
- Eq. 1: right-hand side (sample: 8)
- Eq. 2: coefficient of x (sample: -3)
- Eq. 2: coefficient of y (sample: -1)
- Eq. 2: coefficient of z (sample: 2)
- Eq. 2: right-hand side (sample: -11)
- Eq. 3: coefficient of x (sample: -2)
- Eq. 3: coefficient of y (sample: 1)
- Eq. 3: coefficient of z (sample: 2)
- Eq. 3: right-hand side (sample: -3)
- Press Solve System (results also refresh as you type). The result card shows the Solution, along with x, y, z and Determinant.
Formula used
x = Dₓ ÷ D, y = D_y ÷ D, z = D_z ÷ D
- D = the determinant of the coefficient matrix
- Dₓ = the same determinant with the x column replaced by the right-hand sides (similarly D_y and D_z)
If D = 0 the equations are either dependent (infinitely many solutions) or inconsistent (no solution), and the solver says so instead of dividing by zero.
Worked example
These are the sample values already filled in the calculator above, and the result it returns for them:
| Input | Sample value |
|---|---|
| Number of Unknowns | 3 (x, y, z) |
| Eq. 1: coefficient of x | 2 |
| Eq. 1: coefficient of y | 1 |
| Eq. 1: coefficient of z | -1 |
| Eq. 1: right-hand side | 8 |
| Eq. 2: coefficient of x | -3 |
| Eq. 2: coefficient of y | -1 |
| Eq. 2: coefficient of z | 2 |
| Result | Value |
|---|---|
| Solution | x = 2, y = 3, z = -1 |
| x | 2 |
| y | 3 |
| z | -1 |
| Determinant | -1 |
Tips for using the Linear Equation System Solver
- Write every equation as ax + by + cz = d before entering it, and use 0 for a missing variable.
- Substitute the answer back into each equation to check it.
- For a 2×2 system, the z coefficients and the third equation are ignored.