Question

Solve the system of linear equations using the matrix method.

5x + 2y = 4

7x + 3y = 5

Answer 1

The given equation,

5x + 2y = 4
7x + 3y = 5

A = `[(5,2),(7,3)]`, X = `[(x),(y)]` and B = `[(4),(5)]`

⇒ AX = B

⇒ X = A⁻¹B

The cofactors of the elements of matrix A are as follows:

A₁₁ = 3, A₁₂ = −7, A₂₁ = −2, A₂₂ = 5

Matrix composed of the elements of the cofactor of A = `[(3,-7),(-2,5)]`

adj A = `[(3,-7),(-2,5)] = [(3,-2),(-7,5)]`

|A| = `|(5,2),(7,3)|`

= 15 − 14

= 1 ≠ 0

∴ A⁻¹ = `1/|A|` (adj A)

= `1/1 [(3,-2),(-7,5)]`

X = A⁻¹B = `[(3,-2),(-7,5)][(4),(5)]`

= `[(12 - 10),(-28 + 25)]`

= `[(2),(-3)]`

⇒ `[(x),(y)] = [(2),(-3)]`

⇒ x = 2 and y = −3