If the system of equations
$2x+3y-z=5$
$x+\alpha y+3z=-4$
$3x-y+\beta z=7$
has infinitely many solutions, then $13\alpha\beta$ is equal to
Step-by-Step Solution
Key Concept: For infinitely many solutions, express the first plane as a linear combination of the other two (family of planes method): $2x+3y-z-5=k_1(x+\alpha y+3z+4)+k_2(3x-y+\beta z-7)$. Match coefficients to get equations in $k_1,k_2,\alpha,\beta$, then compute $13\alpha\beta$.
Express $2x+3y-z-5=k_1(x+\alpha y+3z+4)+k_2(3x-y+\beta z-7)$. Matching coefficients: $2=k_1+3k_2$, $3=k_1\alpha-k_2$, $-1=3k_1+\beta k_2$, $-5=4k_1-7k_2$. Solving: $k_2=\frac{13}{19}$, $k_1=\frac{-1}{19}$, $\alpha=-70$, $\beta=\frac{-16}{13}$. Thus $13\alpha\beta=13(-70)\left(\frac{-16}{13}\right)=1120$.
Correct Answer: 2