Question

Find k, if the following equations are consistent:

(k – 1)x + (k – 1)y = 17, (k – 1)x + (k – 2)y = 18, x + y = 5

Answer 1

Given equations are
(k – 1)x + (k – 1)y = 17
(k – 1)x + (k – 2)y = 18
x + y = 5

(k - 1)x + (k - 1)y - 17 = 0
(k - 1)x + (k - 1)y - 18 = 0
x + y - 5 = 0

Since, these equations are consistent.

∴ `|("k"- 1, "k" - 1, -17),("k" - 1, "k" - 2, -18),(1, 1, -5)|` = 0

Applying R₁ → R₁ – R₂, R₂ → R₂ + R₃ we get

⇒`|(0, 1, 1),("k", "k"-1, -23),(1, 1, -5)| = 0`

⇒ 0 (-5k + 5 + 23) - 1 (-5k + 23) +1 [k (k - )] = 0

⇒ 0 (-5k + 28) -1 (-5k + 23) + 1 (k - k + 1) = 0

⇒ 0 + 5k - 23 + k - k + 1

⇒ 5k - 22 = 0

∴  5k = 22 = 0

∴ `"k" = 22/5`