Question

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

`(-2)/x - 1/y - 3/z = 3, 2/x - 3/y + 1/z = -13 and 2/x - 3/z` = – 11

Answer 1

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

Then the given equations become,
– 2p – q – 3r = 3

2p – 3q + r = – 13

2p – 3r = – 11

∴ D = `|(-2, -1, -3),(2, -3, 1),(2, 0, -3)|`

= –2(9 – 0) + 1( –6 – 2) – 3(0 + 6)

= –18 – 8 – 18

= – 44 ≠ 0

D_p = `|(3, -1, -3),(-13, -3, 1),(-11, 0, -3)|`

= 3(9 – 0) + 1(39 + 11) – 3(0 – 33)

= 27 + 50 + 99

= 176

D_q = `|(-2, 3, -3),(2, -13, 1),(2, -11, -3)|`

= – 2(39 + 11) – 3(– 6 – 2) – 3(– 22 + 26)

= – 100 + 24 – 12

= – 88

D_r = `|(-2, -1, 3),(2, -3, -13),(2, 0, -11)|`

= – 2(33 + 0) + 1 (–22 + 26) + 3(0 + 6)

= – 66 + 4 + 18

= – 44

∴ by Cramer's rule,

p = `"D"_"p"/"D" = 176/(-44)` = – 4

q = `"D"_"q"/"D" = (-88)/(-44)` = 2

r = `"D"_"r"/"D" = (-44)/(-44)` = 1

∴ p = `1/x = -4, "q" = 1/y = 2, r = 1/z` = 1

∴ x = `-1/4, y = 1/2,` z = 1.