Systems of Equations Solver (3 Variables)

A system of linear equations is a set of two or more equations that share the same variables. The goal is to find the values of those variables that satisfy all equations simultaneously.

Standard form (2×2 system): a₁x + b₁y = c₁ a₂x + b₂y = c₂

Three main solution methods:

1. Substitution Method Solve one equation for one variable, then substitute into the other.

Step 1: From equation 1, isolate x: x = (c₁ − b₁y) ÷ a₁ Step 2: Substitute into equation 2 and solve for y. Step 3: Substitute y back to find x.

2. Elimination Method (Addition Method) Multiply equations to make coefficients of one variable equal, then add or subtract to eliminate it.

3. Cramer’s Rule (determinant method): x = (c₁b₂ − c₂b₁) ÷ (a₁b₂ − a₂b₁) y = (a₁c₂ − a₂c₁) ÷ (a₁b₂ − a₂b₁)

The denominator (a₁b₂ − a₂b₁) is the determinant of the coefficient matrix. If it equals 0, the system has either no solution (parallel lines) or infinite solutions (same line).

What each variable means:

• a₁, a₂ — coefficients of x in equations 1 and 2
• b₁, b₂ — coefficients of y in equations 1 and 2
• c₁, c₂ — constants on the right-hand side
• Determinant — measures whether the system has a unique solution

Worked example: Solve: 3x + 2y = 16 and x − y = 2

Using substitution: From equation 2, x = y + 2. Substitute: 3(y + 2) + 2y = 16 → 3y + 6 + 2y = 16 → 5y = 10 → y = 2 Then x = 2 + 2 = x = 4

Verify: 3(4) + 2(2) = 12 + 4 = 16 ✓ and 4 − 2 = 2 ✓

Geometric interpretation: Each equation represents a straight line. The solution is the point where the lines intersect. Parallel lines → no solution. Identical lines → infinite solutions.

Real-world applications: Mixture problems (combining solutions of different concentrations), break-even analysis, circuit analysis using Kirchhoff’s laws, and supply/demand equilibrium in economics.