System of Equations Solver (2x2)
Solve two linear equations in two unknowns (x and y) using Cramer’s rule.
- 1
Determinant
a₁·b₂ − a₂·b₁ = 2·-1 − 1·3 = -5Non-zero determinant confirms a unique solution exists. - 2
x numerator (c₁b₂ − c₂b₁)
8·-1 − -1·3 = -5 - 3
x = num_x ÷ det
-5 ÷ -5 = 1 - 4
y numerator (a₁c₂ − a₂c₁)
2·-1 − 1·8 = -10 - 5
y = num_y ÷ det
-10 ÷ -5 = 2
Formula
How this is calculated
You enter six numbers describing two straight lines written as a₁x + b₁y = c₁ and a₂x + b₂y = c₂. Here a₁ and a₂ are the coefficients of x, b₁ and b₂ are the coefficients of y, and c₁ and c₂ are the constants on the right side. All values are plain dimensionless numbers, and decimals or negatives are allowed.
The solver applies Cramer's rule. It first computes the determinant det = a₁b₂ − a₂b₁, which measures whether the two lines actually cross at a single point. It then divides two related cross products by this determinant: x = (c₁b₂ − c₂b₁) ÷ det and y = (a₁c₂ − a₂c₁) ÷ det. The result is the exact intersection point of the two lines.
The key assumption is that the determinant is non-zero. When det = 0 the lines are parallel (no solution) or identical (infinitely many solutions), so no unique x and y exists and the tool flags this instead of dividing by zero. Results are shown to four decimal places, so they are rounded for display.
Examples
| Input | Result |
|---|---|
| 2x + 3y = 8; x − y = −1 | x = 1, y = 2 |
About this calculator
A 2x2 system of linear equations consists of two equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂. Cramer’s rule solves it using the determinant det = a₁b₂ − a₂b₁. When det is non-zero the system has exactly one solution given by x = (c₁b₂ − c₂b₁) ÷ det and y = (a₁c₂ − a₂c₁) ÷ det.
When the determinant equals zero the two lines are either parallel (no solution) or the same line (infinitely many solutions), so no single (x, y) pair can be returned. Geometrically, the solution is the point where the two lines intersect.
Frequently asked questions
Cramer’s rule expresses each unknown as a ratio of determinants. For a 2x2 system it gives x and y directly from the coefficients and constants without elimination.
The determinant tells you whether a unique solution exists. If it is non-zero the lines cross at one point; if it is zero they are parallel or identical and there is no single solution.
Rearrange each equation into the form ax + by = c, then enter the six coefficients a₁, b₁, c₁, a₂, b₂, c₂ in the matching fields.
Also known as
TG we-Calculate Editorial Team. (2026). System of Equations Solver (2x2) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/system-of-equations-calculator
TG we-Calculate Editorial Team. "System of Equations Solver (2x2)." TG we-Calculate. 2026. https://we-calculate.com/calculator/system-of-equations-calculator.
TG we-Calculate Editorial Team, "System of Equations Solver (2x2)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/system-of-equations-calculator
@misc{wecalculate_system_of_equations_calculator, title = {System of Equations Solver (2x2)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/system-of-equations-calculator}}, year = {2026}, note = {TG we-Calculate} }
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