Question

Solve the following linear equations by using Cramer’s Rule:

x + y + z = 6, x – y + z = 2, x + 2y – z = 2

Answer 1

Given equations are
x + y + z = 6,

x – y + z = 2,

x + 2y – z = 2

D = `|(1, 1, 1),(1, -1, 1),(1, 2, -1)|`

= 1(1 – 2) – 1(–1 – 1) + 1(2 + 1)

= 1 (–1) –1 (–2) + 1(3)

= –1 + 2 + 3

= 4 ≠ 0

D_x = `|(6, 1, 1),(2, -1, 1),(2, 2, -1)|`

= 6(1 – 2) – 1(–2 – 2) + 1(4 + 2)

= 6(– 1) – 1 (– 4) + 1(6)

= –6 + 4 + 6

= 4

D_y = `|(1, 6, 1),(1, 2, 1),(1, 2, -1)|`

= 1(–2 – 2) – 6(–1 – 1) + 1(2 – 2)

= 1(–4) – 6(–2) + 1(0)

= –4 + 12 + 0

= 8

D_z = `|(1, 1, 6),(1, -1, 2),(1, 2, 2)|`

= 1(–2 – 4) – 1(2 – 2) + 6(2 + 1)

= 1(– 6) – 1(0) + 6(3)

= –6 + 0 + 18

= 12

By Cramer’s Rule,

x = `"D"_x/"D" = 4/4 = 1,  y = "D"_y/"D" = 8/4` = 2 and

z = `"D"_z/"D" = 12/4` = 3

∴ x = 1, y = 2 and z = 3 are the solutions of the given equations.