System of Equations Solver

a₁

b₁

c₁

a₂

b₂

c₂

Result

x2
y3
Determinant-14
How the lines relateПрямые пересекаются в одной точке
Check, first equation2 × 2 + 3 × 3 = 13
Check, second equation4 × 2 + -1 × 3 = 5

Cramer's rule, with the check printed

Solves a two-unknown system by Cramer's rule, reports the determinant and checks the answer in both equations.

A two-equation system can be solved by substitution, elimination or Cramer's rule. The last suits a calculator best: one formula through determinants, with no branching and no decisions about what to multiply by what.

The determinant also answers the question of how many solutions exist. Non-zero and the lines cross at one point. Zero and they are either parallel, with no solution, or identical, with infinitely many.

What this calculator shows

The values of x and y and the system's determinant
How the lines relate: crossing, parallel or coincident
A substitution check in both equations

What to keep in mind

Equations must be in the form a·x + b·y = c; move unknowns left and constants right.
The determinant is a₁b₂ − a₂b₁ — the same as the determinant of the coefficient matrix.

FAQs

What is Cramer's rule?

Solving through determinants: x is the determinant with the first column replaced, divided by the main determinant. For a 2×2 system it is the fastest route.

What does a zero determinant mean?

The lines are parallel or identical. The first has no solutions, the second infinitely many — the calculator says which.

How do I rearrange a system into this form?

Move every term with an unknown to the left and every number to the right. 3x = 2y + 7 becomes 3x − 2y = 7, so a = 3, b = −2, c = 7.

Does this work for three unknowns?

Cramer's rule generalises but needs four determinants, which is tedious by hand. Gaussian elimination is the usual choice beyond two unknowns.

Worked example

2x + 3y = 13 and 4x − y = 5

Input: a₁=2, b₁=3, c₁=13, a₂=4, b₂=−1, c₂=5

Output: x = 2, y = 3

Note: The determinant is −14, so the solution is unique. Check: 2×2 + 3×3 = 13 and 4×2 − 3 = 5.

Useful links

All tools in Math & Statistics

System of Equations Solver

Result

Related tools

Linear Equation Solver

Matrix Calculator

Quadratic Equation Solver

Useful links