Question

Solve the following equations using Cramer’s Rule:

x + 2y – z = 5, 2x – y + z = 1, 3x + 3y = 8

Answer 1

Given equations are
x + 2y – z = 5
2x – y + z = 1
3x + 3y = 8
i.e. 3x + 3y + 0z = 8

∴ D = `|(1, 2, -1),(2, -1, 1),(3, 3, 0)|`

= 1(0 – 3) – 2(0 – 3) – 1(6 + 3)

= 1(– 3) + 2(3) – 1(9)

= – 3 + 6 – 9

= – 6

D_x = `|(5, 2, -1),(1, -1, 1),(8, 3, 0)|`

= 5(0 – 3) – 2(0 – 8) + (– 1)(3 + 8)

= 5(– 3) + 2(8) + (– 1)(11)
= – 15 + 16 – 11
= – 10

D_y = `|(1, 5, -1),(2, 1, 1),(3, 8, 0)|`

= 1(0 – 8) – 5(0 – 3) + 1(16 – 3)

= 1(–8) + 5(3) + 1(13)
= – 8 + 15 – 13
= – 6

D_z = `|(1, 2, 5),(2, -1, 1),(3, 3, 8)|`

= 1(– 8 – 3) – 2(16 – 3) + 5(6 + 3)

= 1(–11) – 2(13) + 5(9)
= – 11 – 26 + 45
= 8

By Cramer’s Rule,

x = `"D"_x/"D" = (-10)/(-6) = 5/3`

y = `"D"_y/"D" = (-6)/(-6)` = 1

z = `"D"_z/"D" = 8/(-6) = (-4)/3`

∴ x = `5/3, y = 1 and z = (-4)/3` are the solution of the given equations.