If for some α and β in R, the intersection of the following three planes
\[x+4y-2z=1,\qquad x+7y-5z=\beta,\qquad x+5y+\alpha z=5\]
is a line in R3, then α+β is equal to:
Solutions · 1
Official solution
For the system to represent a line, the coefficient determinant vanishes:
\[\begin{vmatrix}1&4&-2\\1&7&-5\\1&5&\alpha\end{vmatrix}=0 \implies (7\alpha+25)-4(\alpha+5)+(-2)(5-7)=0\implies 3\alpha+9=0\implies \alpha=-3.\]
Also the determinant with the constants column (replacing z-column) must vanish:
\[\begin{vmatrix}1&4&1\\1&7&\beta\\1&5&5\end{vmatrix}=0\implies (35-5\beta)-4(5-\beta)+(5-7)=0\implies 13-\beta=0\implies\beta=13.\]
\[\alpha+\beta=-3+13=10.\]