If the system of linear equations 2x−3y=γ+5,αx+5y=β+1 where α,β,γ∈R has infinitely many solutions then the value of 9α+3β+5γ is equal to ______

Given system of equations has infinite many solutions a1a2=b1b2=c1c22α=−35=γ+5β+1⇒3α=−10 and −3β−3=5r+253β+5r=−28Now 9α+3β+5r=−30−28=58

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

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answer is 58.

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Given system of equations has infinite many solutions $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$

$\frac{2}{α} = \frac{- 3}{5} = \frac{γ + 5}{β + 1} \Rightarrow$ $3 α = - 10$ and $- 3 β - 3 = 5 r + 25$

$3 β + 5 r = - 28$

Now $|9 α + 3 β + 5 r| = |- 30 - 28| = 58$

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