Question

Solve the following equations using Cramer’s Rule:

2x – y + 6z = 10, 3x + 4y – 5z = 11, 8x – 7y – 9z = 12

Answer 1

Given equations are
2x – y + 6z = 10
3x + 4y – 5z = 11
8x – 7y – 9z = 12

D = `|(2, -1, 6),(3, 4, -5),(8, -7, -9)|`

= 2(– 36 – 35) – (– 1)(– 27 + 40) + 6(– 21 –32)
= – 142 + 13 – 318
= – 447

D_x = `|(10, -1, 6),(11, 4, -5),(12, -7, -9)|`

= 10(– 36 – 35) – (– 1)(– 99 + 60) + 6(– 77 –48)
= – 710 – 39 – 750
= – 1499

D_y = `|(2, 10, 6),(3, 11, -5),(8, 12, -9)|`

= 2(– 99 + 60) – 10(– 27 + 40) + 6(36 – 88)
= – 78 – 130 – 312
= – 520

D_z = `|(2, -1, 10),(3, 4, 11),(8, -7, 12)|`

= 2(48 + 77) – (– 1)(36 – 88) + 10(– 21 – 32)
= 250 – 52 – 530
= – 332
By Cramer’s Rule,

x = `"D"_x/"D" = (-1499)/(-447) = 1499/447`

y = `"D"_y/"D" = (-520)/(-447) = 520/447`

z = `"D"_z/"D" = (-332)/(-447) = 332/447`

∴ x = `1449/447, y = 520/447 and z = 332/447` are the solutions of the given equations.