Math & Algebra Tool

System of Equations Solver

Solve a system of two linear equations.

x + y =
x + y =

Σ The Formula

Using Cramer's Rule: x = (c₁b₂ - c₂b₁)/(a₁b₂ - a₂b₁) | y = (a₁c₂ - a₂c₁)/(a₁b₂ - a₂b₁)

Real World Examples

Intersection Point
2x + y = 5, x - y = 1 → x = 2, y = 1
Supply & Demand
3x + 2y = 12, x + y = 5 → x = 2, y = 3
Age Problem
x + y = 50, x - y = 10 → x = 30, y = 20
Mixture Problem
0.1x + 0.2y = 15, x + y = 100 → x = 50, y = 50

# About This Calculator

A system of equations is a set of two or more equations with the same variables. Solving the system means finding values that satisfy ALL equations simultaneously. For two linear equations, the solution is typically the point where the lines intersect on a graph.

Systems of equations appear in countless real-world scenarios: economics (supply and demand equilibrium), physics (motion problems), chemistry (mixture problems), business (break-even analysis), and engineering (circuit analysis). They're fundamental to linear algebra and optimization.

This calculator uses Cramer's Rule, an elegant method using determinants. For equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂, it calculates x and y using ratios of determinants. If the determinant is zero, the lines are parallel (no solution) or coincident (infinite solutions).

The calculator automatically detects when no unique solution exists and provides step-by-step calculations. It's perfect for homework, verifying manual solutions, or solving real-world problems involving two unknowns and two constraints.

How To Use

  1. Enter coefficients for both equations.
  2. Click Solve.

Frequently Asked Questions

What does 'no solution' or 'infinite solutions' mean?+

No solution: lines are parallel, never intersect (e.g., x+y=5 and x+y=10). Infinite solutions: lines are identical, overlap completely (e.g., 2x+2y=10 and x+y=5). Most real problems have exactly one solution (lines intersect at one point).

Can I solve systems with more than 2 equations?+

This calculator handles 2 equations with 2 unknowns. For 3+ equations, you need matrix methods (Gaussian elimination, matrix inverse) or specialized software. The principles are similar but calculations become more complex.

How is this used in real life?+

Economics: find equilibrium price/quantity where supply = demand. Business: determine break-even point. Physics: solve motion problems with multiple constraints. Chemistry: balance mixture concentrations. Any situation with two unknowns and two relationships.

What's the difference between substitution and elimination?+

Both solve systems, but differently. Substitution: solve one equation for a variable, substitute into the other. Elimination: add/subtract equations to eliminate a variable. Cramer's Rule (used here) is a third method using determinants - often faster for 2×2 systems.

Can I use this for non-linear equations?+

No, this calculator only handles LINEAR equations (no x², xy, etc.). For non-linear systems (like x²+y²=25 and y=x+1), you need different methods like substitution, graphing, or numerical solvers.

Is System of Equations Solver free to use?+

Yes, System of Equations Solver on Matheric is completely free to use. We believe in accessible education and utility for everyone.

About

A system of equations is a set of two or more equations with the same variables. Solving the system means finding values that satisfy ALL equations simultaneously. For two linear equations, the solution is typically the point where the lines intersect on a graph.

Systems of equations appear in countless real-world scenarios: economics (supply and demand equilibrium), physics (motion problems), chemistry (mixture problems), business (break-even analysis), and engineering (circuit analysis). They're fundamental to linear algebra and optimization.

This calculator uses Cramer's Rule, an elegant method using determinants. For equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂, it calculates x and y using ratios of determinants. If the determinant is zero, the lines are parallel (no solution) or coincident (infinite solutions).

The calculator automatically detects when no unique solution exists and provides step-by-step calculations. It's perfect for homework, verifying manual solutions, or solving real-world problems involving two unknowns and two constraints.

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