Question

Solve the following equations using Cramer’s Rule:

`2/x - 1/y + 3/z = 4, 1/x - 1/y + 1/z = 2, 3/x + 1/y - 1/z ` = 2

Answer 1

Let `1/x = "p", 1/y = "q", 1/z` = r

∴ The given equations become
2p – q + 3r = 4
p – q – r = 2
3p + q – r = 2

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

= 2(1 – 1) – (– 1)(– 1 – 3) + 3(1 + 3)
= 0 – 4 + 12
= 8

D_p = `|(4, -1, 3),(2, -1, 1),(2, 1, -1)|`

= 4(1 – 1) – (– 1)(– 2 –2) + 3(2 + 2)
= 0 – 4 + 12
= 8

D_q = `|(2, 4, 3),(1, 2, 1),(3, 2, -1)|`

= 2(–2 – 2) –4(– 1 – 3) + 3(2 – 6)
= – 8 + 16 – 12
= – 4

D_r = `|(2, -1, 4),(1, -1, 2),(3, 1, 2)|`

= 2(– 2 – 2) – (– 1)(2 – 6) + 4(1 + 3)
= – 8 – 4 + 16
= 4
By Cramer's Rule,

p = `"D"_"p"/"D" = 8/8` = 1

q = `"D"_"q"/"D" = (-4)/8 = (-1)/2`

r = `"D"_"r"/"D" = 4/8 = 1/2`

∴ `1/x = 1, 1/y = (-1)/2, 1/z = 1/2`

∴ x = 1, y = – 2 and z = 2 are the solutions of the given equations.