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System of 2 Linear Equations (Cramer's Rule)

Solve a system of two linear equations (a₁x + b₁y = c₁, a₂x + b₂y = c₂) for x and y using Cramer's Rule determinants.

Linear System: a₁x + b₁y = c₁ & a₂x + b₂y = c₂

Variable x
5.0000
Variable y
1.0000
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Calculation Guide & Reference

Cramer's Rule for Systems of Two Linear Equations

System of equations calculator solves two linear equations (a₁x + b₁y = c₁, a₂x + b₂y = c₂) for x and y using Cramer's Rule determinants.

Standardized Mathematical Formula
x = (c₁b₂ − c₂b₁) / D | y = (a₁c₂ − a₂c₁) / D | D = a₁b₂ − a₂b₁

Cramer's Rule solves a 2×2 linear system by replacing columns of the coefficient matrix with the constants column and taking ratios of determinants — no substitution or elimination steps required.

Variables:
D:Main determinant of the coefficient matrix: a₁b₂ − a₂b₁
a₁,b₁,c₁:Coefficients and constant of the first equation
a₂,b₂,c₂:Coefficients and constant of the second equation
How It Works (Step-by-Step)
  • 1Enter the coefficients a₁, b₁, c₁ for the first equation and a₂, b₂, c₂ for the second.
  • 2View the solved x and y values, or a flag if the system has no unique solution (parallel or coincident lines).
Real-World Numerical Example
Solving 2x + 3y = 13 and x − y = 4

Find x and y for the system 2x + 3y = 13 and x − y = 4.

•Main determinant: D = (2 × −1) − (1 × 3) = −2 − 3 = −5.
•x = (13 × −1 − 4 × 3) ÷ −5 = (−13 − 12) ÷ −5 = −25 ÷ −5 = 5.
•y = (2 × 4 − 1 × 13) ÷ −5 = (8 − 13) ÷ −5 = −5 ÷ −5 = 1.
Result: x = 5 and y = 1. Check: 2(5) + 3(1) = 13 ✓ and 5 − 1 = 4 ✓.
Calculation Best Practices & Tips
Examine the discriminant (b² - 4ac) before solving quadratics to instantly know if roots are real, distinct, repeated, or complex.
In statistical analysis, always check both the mean and median to identify if data distribution is skewed by outliers.
When calculating percentage changes, ensure you divide by the original starting baseline value: ((New - Old) / Old) * 100.

Frequently Asked Questions (FAQ)

A method for solving linear systems using determinants: each variable equals the determinant of the coefficient matrix with that variable's column replaced by the constants, divided by the main determinant.

A zero determinant means the two equations represent parallel lines (no intersection, no solution) or the exact same line (infinitely many solutions) — there is no single unique (x, y) answer.

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