Question

Solve the following equations using Cramer’s Rule:

11x – y – z = 31, x – 6y + 2z = –26, x + 2y – 7z = –24

Answer 1

Given equations are
11x – y – z = 31
x – 6y + 2z = –26
x + 2y – 7z = –24

D = `|(11, -1, -1),(1, -6, 2),(1, 2, -7)|`

= 11(42 – 4) – (– 1)(– 7 –2) + (– 1)(2 + 6)
= 418 – 9 – 8
= 401

D_x = `|(31, -1, -1),(-26, -6, 2),(-24, 2, -7)|`

= 31(42 – 4) – (– 1)(182 + 48) + (– 1)(–52 – 144)
= 1178 + 230 + 196
= 1604

D_y = `|(11, 31, -1),(1, -26, 2),(1, -24, -7)|`

= 11(182 + 48) – 31(– 7 – 2) + (– 1)(– 24 + 26)
= 2530 + 279 – 2
= 2807

D_z = `|(11, -1, 31),(1, -6, -26),(1, 2, -24)|`

= 11(144 + 52) – (– 1)(– 24 + 26) + 31(2 + 6)
= 2156 + 2 + 248
= 2406
By Cramer's Rule,

x = `"D"_x/"D" = 1604/401` = 4

y = `"D"_y/"D" = 2807/401` = 7

z = `"D"_z/"D" = 2406/401` = 6

∴ x = 4, y = 7 and z = 6 are the solutions of the given equations.