Question

Find k if the following equations are consistent:

kx + 3y + 1 = 0, x + 2y + 1 = 0, x + y = 0

Answer 1

Given equations are
kx + 3y + 1 = 0,

x + 2y + 1 = 0,

x + y = 0, i.e., x + y + 0 = 0.

Since these equations are consistent,

${\displaystyle \left|\begin{array}{ccc}\text{k}& 3& 1\\ 1& 2& 1\\ 1& 1& 0\end{array}\right|}$ = 0

∴ k(0 – 1) – 3(0 – 1) + 1(1 – 2) = 0

∴ k(–1) – 3(–1) + 1(–1) = 0

∴ – k + 3 – 1 = 0

∴ k = 2