System of Equations Calculator

Solve a 2×2 system of linear equations using Cramer's rule.

How the Math Works

This calculator uses Cramer's rule to solve systems of two linear equations in the form a₁x + b₁y = c₁ and a₂x + b₂y = c₂. First, it calculates the determinant D = a₁b₂ - a₂b₁. If D ≠ 0, unique solutions exist. The values Dₓ and D_y are found by replacing the appropriate coefficient columns with the constants: Dₓ = c₁b₂ - c₂b₁ and D_y = a₁c₂ - a₂c₁. Finally, x = Dₓ/D and y = D_y/D give the intersection point of the two lines.

Practical Applications

To use this calculator, enter the coefficients a₁, b₁, c₁ from your first equation and a₂, b₂, c₂ from your second equation into the corresponding fields. For example, for the system 3x + 2y = 7 and x - y = 1, you would input a₁=3, b₁=2, c₁=7, a₂=1, b₂=-1, c₂=1. Click calculate to instantly receive the values of x and y that satisfy both equations simultaneously.

Day-to-Day Use

This tool helps with everyday problems involving multiple unknowns, such as determining the cost of items when given total purchases, calculating break-even points for business expenses, or finding the right mixture of ingredients with specific constraints. Instead of manual algebra, you can quickly verify solutions for homework problems or make informed decisions about resource allocation, budgeting, and planning.

Worked example

2x + 3y = 6 and x − y = −2 → x = 0, y = 2.

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