System of Equations Calculator
Enter two linear equations, a₁x + b₁y = c₁ and a₂x + b₂y = c₂, and get the (x, y) solution with elimination steps, or a clear verdict of no/infinite solutions.
How to use this calculator
Two equations, two unknowns. Type all six coefficients and press Calculate, the solver uses elimination (via Cramer’s rule) and checks the answer by substituting back into both equations.
How it works
Elimination in one line: compute the determinant D = a₁b₂ − a₂b₁. If D ≠ 0, x = (c₁b₂ − c₂b₁)/D and y = (a₁c₂ − a₂c₁)/D. If D = 0, the lines are parallel (no solution) or identical (infinitely many), the calculator tells you which.
Worked example
Example: 2x + 3y = 8 and x − y = 1
Step 1, Determinant: D = (2)(−1) − (1)(3) = −5.
Step 2, x: x = ((8)(−1) − (1)(3))/(−5) = (−11)/(−5) = 2.2.
Step 3, y: y = ((2)(1) − (1)(8))/(−5) = (−6)/(−5) = 1.2.
Step 4, Check: 2(2.2)+3(1.2) = 8 ✓; 2.2 − 1.2 = 1 ✓.
Common mistakes
- Ignoring the D = 0 case: a zero determinant is not an error, it means the lines are parallel or the same line. Read the verdict instead of forcing an answer.
- Sign slips in the cross-products: (c₁b₂ − c₂b₁) has a minus in the middle; copying it as plus is the classic error.
- Skipping the check: one substitution catches nearly every arithmetic slip. Always verify in both equations.
When to use it
Break-even points (cost = revenue), mixture problems, intersection of two lines, supply-and-demand equilibrium, and any word problem with two unknown quantities and two independent conditions. Unsure which method to use by hand? See elimination vs substitution.
Working with three unknowns? See solving a 3×3 system worked end to end by elimination.
Frequently asked questions
By elimination or substitution. Elimination (used here): combine the equations to cancel one variable, solve for the other, then back-substitute.
The two lines are parallel and distinct, they never meet, so no (x, y) satisfies both. The determinant D is zero.
The two equations describe the same line. Every point on the line satisfies both, the system is dependent, not contradictory.
Yes – 3×3 systems are now live. Switch System size to 3 equations, 3 unknowns to solve three equations in three unknowns, with step-by-step elimination and checks.