Question

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

x + z = 1, y + z = 1, x + y = 4

Answer 1

Given equations are
x + z = 1, i.e., x + 0y + z = 1,

y + z = 1, i.e., 0x + y + z = 1,

x + y = 4, i.e., x + y + 0z = 4.

D = `|(1, 0, 1),(0, 1, 1),(1, 1, 0)|`

= 1(0 – 1) – 0 + 1(0 – 1)

= 1(–1) + 1(–1)

= –1 – 1

= –2 ≠ 0

D_x = `|(1, 0, 1),(1, 1, 1),(4, 1, 0)|`

= 1(0 – 1) – 0 + 1(1 – 4)

= 1(–1) + 1(–3)

= –1 – 3

= –4

D_y = `|(1, 1, 1),(0, 1, 1),(1, 4, 0)|`

= 1(0 – 4) – 1(0 – 1) + 1(0 – 1)

= 1(–4) – 1(–1) + 1(–1)

= –4 + 1 – 1

= –4

D_z = `|(1, 0, 1),(0, 1, 1),(1, 1, 4)|`

= 1(4 – 1) – 0 + 1(0 – 1)

= 1(3) + 1(–1)

= 3 – 1

= 2

By Cramer’s Rule,

x = `"D"_x/"D" = (-4)/(-2) = 2, y = "D"_y/"D" = (-4)/(-2)` = 2,

z = `"D"_z/"D" = 2/(-2)` = – 1

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