Question

Find the value of k for which the system of equations has a unique solution:
5x – 7y = 5,
2x + ky = 1.

Answer 1

The given system of equations are
5x - 7y – 5 = 0               ….(i)
2x + ky - 1 = 0              …(ii)
This system is of the form:
${\displaystyle a_{1}x+b_{1}y+c_1=0}$
${\displaystyle a_{2}x+b_{2}y+c_2=0}$
where,${\displaystyle a_1=5,b_1=-7,c_1=-5\text{and}{a}_2=2,b_2=k,c_2=-1}$
Now, for the given system of equations to have a unique solution, we must have:
${\displaystyle \frac{a_1}{a_2}\ne \frac{b_1}{b_2}}$

${\displaystyle \Rightarrow \frac{5}{2}\ne \frac{-7}{k}}$
${\displaystyle \Rightarrow k \ne -\frac{14}{5}}$
Hence${\displaystyle k\ne -\frac{14}{5}}$