Linear Combination Calculator

The linear combination method—often called the addition or elimination method—solves a system of two linear equations in two variables by combining them to eliminate one variable. Given two standard equations: a₁x + b₁y = c₁ and a₂x + b₂y = c₂, we first compute the primary determinant using Cramer's rule formula: Determinant = a₁b₂ - b₁a₂. If the determinant is non-zero, unique solutions for x and y exist. The tool computes specific sub-determinants (Determinant X and Determinant Y) to determine the exact coordinates, utilizing division: x = Determinant X / Determinant and y = Determinant Y / Determinant.

Let us solve the system of equations: 5x + 2y = 12 and 8x + 12y = 28. First, identify our coefficients: a₁ = 5, b₁ = 2, c₁ = 12, and a₂ = 8, b₂ = 12, c₂ = 28. Next, calculate the primary determinant: (5 × 12) - (2 × 8) = 60 - 16 = 44. Then, calculate Determinant X: (12 × 12) - (2 × 28) = 144 - 56 = 88. Finally, calculate Determinant Y: (5 × 28) - (12 × 8) = 140 - 96 = 44. Dividing these gives our final coordinates: x = 88 / 44 = 2, and y = 44 / 44 = 1. The solution is x = 2, y = 1.

To get the most out of this calculator and avoid common algebraic pitfalls, keep these guidelines in mind:

• Check for Consistency: If your determinant evaluates to zero, the system either has infinitely many solutions or no solution at all.
• Standard Form: Ensure your input equations are properly arranged in standard form (Ax + By = C) before entering the coefficients.
• Verify Fractions: When decimals appear in the output, cross-verify by substituting the resulting values back into your original equations.