System of Equations Solver

Written byOleh Soloninupdated 5 October 2026

a₁

b₁

c₁

a₂

b₂

c₂

Result

x2
y3
Step-by-step solution
By substitution From the second equation: y = 4x − 5 Substitute into the first: 2x + 3(4x − 5) = 13 14x − 15 = 13 14x = 28 x = 2 y = 4 · 2 − 5 = 3
Determinant-14
How the lines relateThe lines meet at one point
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.

How to use

Enter the coefficients of both equations.
Read x and y with the substitution worked through.
Read the determinant: it tells you whether the lines cross, run parallel or coincide.

Three outcomes, not one

The system 2x + 3y = 13, 4x − y = 5 has a single answer: x = 2, y = 3. That happens whenever the determinant is non-zero — here it is −14, and the two lines cross at one point.

A zero determinant means parallel lines: either no solution at all, or infinitely many if both equations describe the same line. The calculator names which case it is instead of dividing by zero.

The determinant is a₁b₂ − a₂b₁; in the default example it is −14.
Determinant	Geometry	Solutions
non-zero	lines cross	exactly one
zero, coefficients proportional but constants not	parallel lines	none
zero, constants proportional too	the same line	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.

How is 2x + 3y = 13 with 4x − y = 5 solved?

x = 2 and y = 3. Rearranging the second equation gives y = 4x − 5; substituting it into the first leaves 14x = 28, and both equations check out.

When does a system have no solution?

When the determinant is zero and the equations contradict each other — the lines are parallel and never meet. If instead they describe the same line, every point on it is a solution.

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.

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System of Equations Solver

Result

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