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.

Examples

2x + 3y = 8, x - y = -1

Unique solution: x = 1, y = 2

x coefficient: 2 none
y coefficient: 3 none
constant: 8 none
x coefficient: 1 none
y coefficient: -1 none
constant: -1 none

Solution

x = 1, y = 2

Determinant

-5 none

Solution Type

Unique solution

Solution steps

\det = (2)(-1) - (1)(3) = -5

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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}

Variables, symbols and units

$a_1$: 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

Calculation method explained

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.

Examples

2x + 3y = 8, x - y = -12 none · 3 none → x = 1, y = 2

Unique solution: x = 1, y = 2

x coefficient: 2 none
y coefficient: 3 none
constant: 8 none
x coefficient: 1 none
y coefficient: -1 none
constant: -1 none

Solution: x = 1, y = 2

Parallel lines1 none · 1 none → No solution

x + y = 1 and 2x + 2y = 5 have no solution

x coefficient: 1 none
y coefficient: 1 none
constant: 1 none
x coefficient: 2 none
y coefficient: 2 none
constant: 5 none

Solution: No solution

3x - 2y = 7, x + 4y = 93 none · -2 none → x = 3.285714, y = 1.428571

Finding the intersection point

x coefficient: 3 none
y coefficient: -2 none
constant: 7 none
x coefficient: 1 none
y coefficient: 4 none
constant: 9 none

Solution: x = 3.285714, y = 1.428571

Frequently Asked Questions

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.

Linear System Solver

2x + 3y = 8, x - y = -1

How It Works

Examples

Frequently Asked Questions

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