Systems of Equations Calculator

Solve 2×2 systems of linear equations using substitution or elimination. Find (x, y) that satisfies both equations simultaneously. Handles unique solutions, no solution, and infinite solutions.

∑ Algebra📐 ax + by = c; dx + ey = f🔢 Math
a (x-coeff eq 1)
b (y-coeff eq 1)
c (right side eq 1)
d (x-coeff eq 2)
e (y-coeff eq 2)
f (right side eq 2)
Please enter valid values.

Formula & Reference

VariableSymbolFormulaUnits
Systems of Equations Calculator—ax + by = c; dx + ey = f(x, y)

Step-by-Step Examples

Example 1
Classic System

2x + 3y = 12 and x − y = 1

  • det = 2(−1) − 3(1) = −5
  • x = (12(−1) − 3(1))/(−5) = (−15)/(−5) = 3
  • y = (2(1) − 12(1))/(−5) = (−10)/(−5) = 2
  • Check: 2(3)+3(2)=12 ✓, 3−2=1 ✓
✓ x=3, y=2
Example 2
Supply & Demand

Supply: p = 2q + 10. Demand: p = −q + 40. Find equilibrium.

  • 2q + 10 = −q + 40 → 3q = 30 → q = 10
  • p = 2(10) + 10 = 30
  • Equilibrium: q=10, p=30
✓ q=10, p=30
Example 3
No Unique Solution

2x + 4y = 8 and x + 2y = 5.

  • det = 2(2) − 4(1) = 0
  • Parallel lines: no intersection
  • No solution exists
✓ No unique solution

Real-World Applications

💼
Economics
Supply and demand equilibrium, cost and revenue intersection.
🧪
Chemistry
Mixture problems: two solutions combined to reach target concentration.
📊
Finance
Two unknowns: interest rate and principal, solve simultaneously.
🏫
Education
Core algebra skill: systems appear in every standardized test.

Common Mistakes to Avoid

⚠️
Wrong sign on determinant

Cramer's rule: det = ae − bd. Common error: computing ad − bc instead.

⚠️
Not checking the solution

Always substitute back into BOTH equations. One check is not enough.

⚠️
Assuming unique solution always exists

When det = 0, use the original equations to determine if parallel (no solution) or same line (infinite solutions).

Frequently Asked Questions

What is a system of equations? ▾
Two or more equations with the same variables. The solution is the point(s) where all equations are satisfied simultaneously.
What are the solution methods? ▾
Substitution: solve one equation for a variable, substitute into other. Elimination: add/subtract multiples of equations to eliminate a variable. Matrix: use Cramer's rule or row reduction.
What does the determinant tell us? ▾
det ≠ 0: unique solution (lines intersect). det = 0: either no solution (parallel lines) or infinite solutions (same line).
How do I solve 3×3 systems? ▾
Use Gaussian elimination (row reduction) or matrix inversion. More complex but same principle: reduce to triangular form, then back-substitute.
What is linear independence? ▾
Two equations are independent if neither is a multiple of the other. Independent equations always give a unique solution (det ≠ 0).

Related Math Calculators

∑ Linear Equation∑ Matrix∑ Determinant∑ Slope∑ Quadratic

Formula Explorer connections

Interpretation: This formula describes an algebraic relationship among variables, functions, equations, roots or sequences. Assumption: Respect the expression’s domain and excluded values. Check roots in the original equation because transformations can introduce extraneous solutions.

Transfer Function Calculator →Transfer Function Poles and Zeros Calculator →Wave Equation Approximation Calculator →Math Formula Explorer →