Question

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

x + y − 2z = –10, 2x + y – 3z = –19, 4x + 6y + z = 2

Answer 1

The given equations are

x +  y – 2z = – 10    ...(1)

2x + y – 3z = – 19   ...(2)

4x + 6y + z = 2     ...(3)

∴ D = `|(1, 1, -2),(2, 1, -3),(4, 6, 1)|`

= 1(1 + 18) – 1(2 + 12) – 2(12 – 4)

= 19 – 14 – 16

= – 11 ≠ 0

D_x = `|(-10, 1, -2),(-19, 1, -3),(2, 6, 1)|`

= – 10(1 + 18) – 1 ( –19 + 6) – 2( –114 – 2)

= – 190 + 13 + 232

= 55

D_y = `|(1, -10, -2),(2, -19, -3),(4, 2, 1)|` 

= 1(– 19 + 6) + 10 (2 + 12) – 2 (4 + 76)

= – 13 + 140 – 160

= – 33 

D_z = `|(1, 1, -10),(2, 1, -19),(4, 6, 2)|`

= 1(2 + 114) – 1 (4 +76) – 10(12 – 4)

= 116 – 80 – 80

= – 44

∴ by Cramer's rule,

x = `"D"_x/"D" = 55/(-11)` = –  5

y = `"D"_y/"D" = (-33)/(-11)` = 3

z = `"D"_z/"D" = (-44)/(-11)` = 4

∴ x = – 5, y = 3, z = 4.