Question

Solve the following equations using Cramer’s Rule:

`1/x + 1/y + 1/z = - 2,  1/x - 2/y + 1/z = 3,  2/x - 1/y + 3/z` = -1

Answer 1

Let `1/x  "p", 1/y = "q", 1/z` = r

∴ The given equations become
p + q + r = – 2
p – 2q + r = 3
2p  – q + 3r = – 1

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

= 1(– 6 + 1) – 1(3 – 2) + 1(– 1 + 4)
= - 5 – 1 + 3
= – 3

D_p = `|(-2, 1, 1),(3, -2, 1),(-1, -1, 3)|`

= – 2(– 6 + 1) –  1(9 + 1) + 1(– 3 – 2)
= 10 – 10 – 5
= – 5

D_q = `|(1, -2, 1),(1, 3, 1),(2, -1, 3)|`

= 1(9 + 1) + 2(3 – 2) + 1(– 1 –6)
= 10 + 2 – 7
= 5

D_r = `|(1, 1, -2),(1, -2, 3),(2, -1, -1)|`

= 1(2 + 3) – 1(– 1 –6) – 2(– 1 + 4)
= 5 + 7 – 6
= 6
By Cramer's Rule,

p = `"D"_"p"/"D" = (-5)/(-3) = 5/3`

q = `"D"_"q"/"D" = (5)/-3 = (-5)/3`,

r = `"D"_"r"/"D" = 6/(-3)` = – 2

∴ `1/x = 5/3, 1/y = (-5)/3, 1/z ` = – 2

∴ x = `3/5, y = (-3)/5, z = (-1)/2` are the solution of the given equatios.