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System of Equations Calculator

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Enter the coefficients for two linear equations (ax + by = c) to solve for x and y.

System of Equations Calculator

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a₁x + b₁y = c₁  and  a₂x + b₂y = c₂

Solution
x = 1, y = 2
Solved with Cramer's rule. If the system has no unique solution, the calculator will say so.

The formula (Cramer's rule)

D = a₁b₂ - b₁a₂ x = (c₁b₂ - b₁c₂) / D y = (a₁c₂ - c₁a₂) / D
Example

2x + 3y = 8 and x - y = -1: D = (2)(-1)-(3)(1) = -5, x = ((8)(-1)-(3)(-1))/-5 = 1, y = ((2)(-1)-(8)(1))/-5 = x=1, y=2.

Step-by-step guide

  1. Write both equations in the form ax + by = c.
  2. Enter the six coefficients.
  3. Read the solution for x and y.

What "no solution" and "infinitely many" actually mean graphically

Each linear equation represents a straight line, and solving the system means finding where those two lines intersect. A single solution means the lines cross at exactly one point - the normal case. "No solution" means the lines are parallel and never meet (same slope, different position). "Infinitely many solutions" means both equations actually describe the exact same line, so every point on it satisfies both equations at once.

Common mistakes

Mixing up which number is a, b, or c - write each equation carefully in ax + by = c form before entering the coefficients, including negative signs.
Forgetting to move a term across the equals sign correctly - if your original equation is "2x = 8 - 3y", it needs to be rewritten as "2x + 3y = 8" before entering the coefficients.

Frequently asked questions

What if D equals zero?

It means the two equations don't have a single unique intersection point - either the lines are parallel (no solution) or they're actually the same line (infinitely many solutions), and the calculator tells you which.

Can this solve three equations with three unknowns?

This version handles two equations with two unknowns - for three variables, the same Cramer's rule approach extends using 3x3 determinants, which you can compute with our Matrix Calculator.

Is Cramer's rule the only way to solve this?

No - substitution and elimination are common hand-solving methods too, and all correct methods arrive at the same answer. Cramer's rule is convenient here because it's easy to compute directly from the six coefficients.

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