If the system of equations \(\begin{aligned}& 2 x+3 y-z=5 \\& x+\alpha y+3 z=-4 \\& 3 x-y+\beta z=7\end{aligned}\) has i…

\(If the system of equations

2x+3y−z=5x+αy+3z=−43x−y+βz=7

has infinitely many solutions, then 13αβ is equal to

\begin{aligned}&2=\mathrm{k}_1+3 \mathrm{k}_2, 3=\mathrm{k}_1 \alpha-\mathrm{k}_2,-1=3 \mathrm{k}_1+\beta \mathrm{k}_2,-5=4 \mathrm{k}_1-7 \mathrm{k}_2\\&\text { On solving we get }\\&& k_2=\frac{13}{19}, k_1=\frac{-1}{19}, \alpha=-70, \beta=\frac{-16}{13} \\& 13 \alpha \beta=13(-70)\left(\frac{-16}{13}\right) \\& =1120\end{aligned}\)