Question

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

Answer 1

Given

$2z+3y=7$

$2ax+(\alpha +\beta )y=28$

for linear equations $a_{1}x+b_{1}y+c_1=0,a_{2}x+b_{2}y+c_2=0$

$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\Rightarrow$for infinite solutions

$2x+3y-7=0$

$2\alpha x+(\alpha +\beta )-28=0$

$a_1=2,b_{=}3,c_1=-7$

$a_2=2\alpha ,b_1=(\alpha +\beta ),c_2=28$

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

$\frac{2}{2\alpha }=\frac{7}{28}$

$\frac{1}{\alpha }=\frac{7}{28}$

$\alpha =\frac{28}{7}=4\therefore \alpha =4$

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

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

$\frac{\alpha }{3}+\frac{\beta }{3}=\frac{28}{3}$

$\frac{\beta }{3}=\frac{28}{7}-\frac{4}{3}$

$=4-\frac{4}{3}$

$\frac{\beta }{\alpha }=\frac{12-4}{3}$

$\beta =8$