Question

Find the value of k for which the system of equations
8x + 5y = 9, kx + 10y = 15
has a non-zero solution.

Answer 1

The given system of equations:
8x + 5y = 9
8x + 5y - 9 = 0                   ….(i)
kx + 10y = 15
kx + 10y - 15 = 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, ${\displaystyle a_1=8,b_1=5,c_1=-9\text{and}{a}_2=k,b_2=10,c_2=–15}$
In order that the given system has no solution, we must have:
${\displaystyle \frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}}$
${\displaystyle i.e., \frac{8}{k}=\frac{5}{10}\ne \frac{-9}{-15}}$
${\displaystyle i.e.,\frac{8}{k}=\frac{1}{2}\ne \frac{3}{5}}$
${\displaystyle \frac{8}{k}=\frac{1}{2}\text{and}\frac{8}{k}\ne \frac{3}{5}}$
${\displaystyle \Rightarrow k=16\text{and}k\ne \frac{40}{3}}$
Hence, the given system of equations has no solutions when k is equal to 16.