Question

Answer the following question:

Solve the following linear equations by Cramer’s Rule:

`1/x + 1/y = 3/2, 1/y + 1/z = 5/6, 1/z + 1/x = 4/3`

Answer 1

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

Then the given equations become,

p + q = `3/2`

q + r = `5/6`

p + r = `4/3`

∴ D = `|(1, 1, 0),(0, 1, 1),(1, 0, 1)|`

= 1(1 – 0) – 1(0 – 1) + 0

= 1 + 1

= 2 ≠ 0

D_p = `|(3/2, 1, 0),(5/6, 1, 1),(4/3, 0, 1)|` 

= `3/2 (1 - 0) - 1(5/6 - 4/3) + 0`

= `3/2 - ((15 - 24)/18)`

= `3/2 + 9/18`

= 2

D_q = `|(1, 3/2, 0),(0, 5/6, 1),(1, 4/3, 1)|`

= `1(5/6 - 4/3) - 3/2(0 - 1) + 0`

= `(15 - 24)/18 + 3/2`

= `-9/18 + 3/2`

= 1

D_r = `|(1, 1, 3/2),(0, 1, 5/6),(1, 0, 4/3)|`

= `1(4/3 - 0)-1(0 - 5/6) + 3/2(0 - 1)`

= `4/3 + 5/6 - 3/2`

= `(8 + 5 - 9)/6`

= `4/6`

= `2/3`

∴ p = `"D"_"p"/"D" = 2/2` = 1

q = `"D"_"q"/"D" = 1/2`

r = `"D"_"r"/"D" = ((2/3))/2 = 1/3`

∴ p = `1/x = 1, "q" = 1/y = 1/2, "r" = 1/z = 1/3`

∴ x = 1, y = 2, z = 3.