Question

Find the values of $\alpha$ and $\beta$ for which the following system of linear equations has infinite number of solutions:
$2x+3y=7$
$2\alpha x+(\alpha +\beta )y=28$.

Answer 1

he given system of equations is
$2x+3y-7=0$
$2\alpha x+(\alpha +\beta )y-28=0$

This system of equation is of the form
$a_{1}x+b_{1}y+c_1=0$
$a_{2}x+b_{2}y+c_2=0$

where $a_1=2,b_1=3,c_1=-7$

and $a_2=2\alpha ,b_2=(\alpha +\beta )$ and $c_2=-28$

For infinitely many solutions, we must have
$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

The given system of equations will have infinite number of solutions, if

$\frac{2}{2\alpha }=\frac{3}{\alpha +\beta }=\frac{-7}{-28}$

$\Rightarrow \frac{1}{\alpha }=\frac{3}{\alpha +\beta }=\frac{1}{4}$

$\Rightarrow \frac{1}{\alpha }=\frac{1}{4}$ and $\frac{3}{\alpha +\beta }=\frac{1}{4}$

$\Rightarrow \alpha =4$ and $\alpha +\beta =12$

$\Rightarrow \alpha =4$ and $\beta =8$

Hence, the given system of equations will have infinitely many solutions, if $\alpha =4$ and $\beta =8$.