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Solving Simultaneous Equations With Matrices

A system of two linear equations can be written as a matrix equation, and solved using the inverse matrix.

Equations as a Matrix

The system ax + by = e, cx + dy = f can be written as a single matrix equation AX = B, where: A = (a c b d), X = (x y), B = (e f) To solve for X, multiply both sides by A^{- 1} : X = A⁻¹B.

Matrix lab

Matrix A

Right-hand side B

Determinant of A: 1.0000

Solution X ≈ [5.000, -2.000]

Change a value and watch:

This matrix equation represents 2 x + y = 8 and 5 x + 3 y = 19 . Solving multiplies the right-hand side by the inverse matrix.

The solution matches Example 1 below exactly.

Example 1 — solving a system

Solve 2 x + y = 8 and 5 x + 3 y = 19 using matrices. 1 A = (2513), B = (819) . From the last lesson, A^{- 1} = (3 - 5 - 1 2) 2 Multiply A^{- 1} by B: top entry = (3 \times 8) + (- 1 \times 19) = 24 - 19 = 5 3 Bottom entry = (- 5 \times 8) + (2 \times 19) = - 40 + 38 = - 2 Answer: x = 5, y = - 2 . Check: 2 (5) + (- 2) = 8 ✓, and 5 (5) + 3 (- 2) = 25 - 6 = 19 ✓

Example 2 — a second system

Solve 3 x + 5 y = 1 and 2 x + 4 y = 2 using matrices. 1 A = (3254), det(A) = 2, so A^{- 1} = (2 - 1 - 2.5 1.5) 2 Multiply A^{- 1} by B = (12) : top entry = (2 \times 1) + (- 2.5 \times 2) = 2 - 5 = - 3 3 Bottom entry = (- 1 \times 1) + (1.5 \times 2) = - 1 + 3 = 2 Answer: x = - 3, y = 2 . Check: 3 (- 3) + 5 (2) = - 9 + 10 = 1 ✓, and 2 (- 3) + 4 (2) = - 6 + 8 = 2 ✓

Practice

Solve 2 x + y = 8 and 5 x + 3 y = 19 (the same system as Example 1). Find x .

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