Question

Find the value of k for which the system of equations
3x - y = 5, 6x - 2y = k
has no solution

Answer 1

The given system of equations:
3x - y - 5 = 0                      ….(i)
And, 6x - 2y + k = 0             ….(ii)
These equations are of the following form:
${\displaystyle a_{1}x+b_{1}y+c_1=0,a_{2}x+b_{2}y+c_2=0}$
where,${\displaystyle a_1=3,b_1=-1,c_1=-5\text{and}{a}_2=6,b_2=-2,c_2=k}$
In order that the given system has no solution, we must have:
${\displaystyle \frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}}$
${\displaystyle i.e.,\frac{3}{6}=\frac{-1}{-2}\ne -\frac{5}{k}}$
${\displaystyle \Rightarrow \frac{-1}{-2}\ne \frac{-5}{k}\Rightarrow k\ne -10}$
Hence, equations (i) and (ii) will have no solution if k ≠ -10.