Question

If the pair of linear equation $5x+ky=-7,x+2y=3$ is inconsistent, then find the value of $k$.

Answer 1

If a pair of linear equations is given by

$a_{1}x+b_{1}y+c_1=0$ and

$a_{2}x+b_{2}y+c_2=0$

When the system of linear equations will represent two Parallel Lines there is no point of intersection and consequently, there is no pair of values of $x$ and $y$ which satisfy both equations. Thus, the system has no solution and such pair of linear equation is inconsistent pair of linear equations

For parallel lines(inconsistent)

$\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$

The given pair of linear equations are

$5x+ky+7=0$ ........$\left(1\right)$

$x+2y-3=0$ .........$\left(2\right)$

On comparing with General form of a pair of linear equations in two variables $x$ & $y$ is:

we get

$a_1=5,b_1=k,c_1=7,a_2=1,b_2=2,c_2=-3$

$\Rightarrow \frac{5}{1}=\frac{k}{2}\ne \frac{7}{-3}$

$\Rightarrow \frac{k}{2}=5$

$\therefore k=10$