Animated Solution for Mathematics - Matrices and Determinants: If for some $\alpha$ and $\beta$ in $\mathbb{R}$, the intersection of the following three planes $x+4y-2z=1$, $x+7y-5z=\beta$, $x+5y+\alpha z=5$ is a line in ${\mathbb{R}}^3$, then $\alpha +\beta$ is equal to:

Imagine standing in a three-dimensional room with three large, flat sheets of glass representing planes in ${\mathbb{R}}^3$. The planes are defined by the following equations:

We are given that these three planes intersect along a single straight line. In linear algebra, this geometric configuration implies that the system of equations has infinitely many solutions and is linearly dependent.

To find the values of $\alpha$ and $\beta$, we utilize the properties of determinants. For a system of three linear equations to possess infinitely many solutions, the main determinant of the coefficient matrix, $\Delta$, must be zero.

Furthermore, for the system to be consistent, the augmented determinants ${\Delta }_x,{\Delta }_y,$ and ${\Delta }_z$ must also vanish. We begin by constructing the coefficient matrix:

Expanding this determinant along the first row, we obtain:

Simplifying the expression step-by-step:

Combining like terms yields:

Solving this linear equation, we find $\alpha =-3$.

To determine $\beta$, we ensure the system remains consistent by setting ${\Delta }_z=0$. We construct ${\Delta }_z$ by replacing the third column of the coefficient matrix with the constants from the right-hand side:

Expanding this along the first row gives:

Simplifying the terms:

Combining the $\beta$ terms and the constants results in:

This leads us directly to $\beta =13$.

We have successfully determined the values $\alpha =-3$ and $\beta =13$. The problem asks for the sum $\alpha +\beta$.

Substituting our values, we calculate:

The final result is $10$.