Question

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

Answer 1

2x+3y=7 --- (1)
$2\alpha \beta x+(\alpha +\beta )y=28$ ---(2)
If equation (2) is a multiple of equation (1)
then there are infinite solutions.
We can see RHS of (2) = 4 rhs of (1).
$\therefore Multiply\left(1\right)by4.$
8x+12y=28
$2\alpha \beta x+(\alpha +\beta )y=y.$
$\therefore 2\alpha \beta =8$
$\alpha +\beta =12$
$\alpha +\frac{4}{\alpha }=12$
${\alpha }^2-12\alpha +4=0$
$\alpha =\frac{12\pm \sqrt{14-16}}{2}$
$=\frac{12\pm \sqrt{128}}{2}$
$=\frac{12+11.31}{2}$
$\frac{23.31}{2}$
=11.65 \&
$\frac{69}{2}=34.5$