CalcLibrary

Linear System Solver

Solve a system of two linear equations (ax + by = c, dx + ey = f) using Cramer's rule. Find x and y, or detect parallel/coincident lines.

Equation 1
Equation 2
Examples

Unique solution: x = 1, y = 2

Solution
x = 1, y = 2
x
1
y
2
Determinant
-5 none
Solution Type
Unique solution
Solution steps
2x + 3y = 8, 1x - 1y = -1, \det = (2)(-1) - (1)(3) = -5, x = \frac{(8)(-1) - (-1)(3)}{-5} = 1, y = \frac{(2)(-1) - (1)(8)}{-5} = 2

For this 2x2 system, x = 1 and y = 2 satisfy both equations at the same time.

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Examples

How It Works

Formula

det⁡=a1b2−a2b1\det = a_1 b_2 - a_2 b_1

x=c1b2−c2b1det⁡x = \frac{c_1 b_2 - c_2 b_1}{\det}

y=a1c2−a2c1det⁡y = \frac{a_1 c_2 - a_2 c_1}{\det}

Variables

a1a_1

Coefficient of x in equation 1

b1b_1

Coefficient of y in equation 1

c1c_1

Constant term of equation 1

a2a_2

Coefficient of x in equation 2

b2b_2

Coefficient of y in equation 2

c2c_2

Constant term of equation 2

Enter coefficients for two equations: a1x + b1y = c1 and a2x + b2y = c2. The solver computes the determinant (a1b2 - a2b1). If non-zero, it applies Cramer's rule. If zero, it checks whether the system is inconsistent or dependent.

Frequently Asked Questions

01What is a system of linear equations?
A system of two linear equations is a pair of equations, each describing a straight line. The solution is the point where the lines intersect.
02What is Cramer's rule?
Cramer's rule uses determinants to solve linear systems. For 2 equations: x = (c1*b2 - c2*b1) / det, y = (a1*c2 - a2*c1) / det, where det = a1*b2 - a2*b1.
03What if the determinant is zero?
A zero determinant means the lines are either parallel (no solution) or coincident (infinitely many solutions). The calculator detects which case applies.
04Can this solve 3 or more equations?
This calculator is designed for 2x2 systems. For larger systems, Gaussian elimination or matrix methods are needed.
05What are parallel vs coincident lines?
Parallel lines never meet (no solution). Coincident lines are the same line (infinitely many solutions). Both cases have a zero determinant.

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