What 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.

What 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.

What 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.

Can this solve 3 or more equations?

This calculator is designed for 2x2 systems. For larger systems, Gaussian elimination or matrix methods are needed.

What 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.

MATMathematics

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.

a (equation 1: ax + by = c)

b (equation 1)

c (equation 1)

d (equation 2: dx + ey = f)

e (equation 2)

f (equation 2)

Try an example

Enter values above to see results

How It Works

Formula

\det = a_1 b_2 - a_2 b_1

x = \frac{c_1 b_2 - c_2 b_1}{\det}

y = \frac{a_1 c_2 - a_2 c_1}{\det}

Where

$a_1$: Coefficient of x in equation 1
$b_1$: Coefficient of y in equation 1
$c_1$: Constant term of equation 1
$a_2$: Coefficient of x in equation 2
$b_2$: Coefficient of y in equation 2
$c_2$: 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.

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