Question

If the system of linear equations
$2 x-3 y=\gamma+5$
$\alpha x +5 y =\beta+1$, where $\alpha, \beta, \gamma \in R$ has infinitely many solutions, then the value of $|9 \alpha+3 \beta+5 \gamma|$ is equal to

Answer 1

$2 x-3 y=\gamma+5$
$\alpha x+5 y=\beta+1$
Infinite many solution
$\frac{\alpha}{2}=\frac{5}{-3}=\frac{\beta+1}{\gamma+5} $
$\alpha=\frac{-10}{3}, \quad 5 \gamma+25=-3 \beta-3 $
$9 \alpha=-30, \quad 3 \beta+5 \gamma=-28 $
Now, $9 \alpha+3 \beta+5 \gamma=-58 $
$|9 \alpha+3 \beta+5 \gamma|=58$