JEE Main · 2022 · 28 Jun 2022 · Shift 2 · Matrices

For a system of two linear equations to have infinitely many solutions, the ratios of coefficients and constants must be equal:
$\frac{\alpha }{2}=\frac{5}{-3}=\frac{\beta +1}{\gamma +5}$.

From $\frac{\alpha }{2}=\frac{5}{-3}$, we get $\alpha =\frac{-10}{3}$, so $9\alpha =-30$.

From $\frac{5}{-3}=\frac{\beta +1}{\gamma +5}$, cross-multiplying gives:
$5(\gamma +5)=-3(\beta +1)$
$5\gamma +25=-3\beta -3$
$3\beta +5\gamma =-28$.

Therefore:
$9\alpha +3\beta +5\gamma =-30+(-28)=-58$.

The absolute value is 58.