Animated Solution for Mathematics - Matrices and Determinants: If the system of equations $x+y+z=6$, $2x+5y+\alpha z=\beta$, $x+2y+3z=14$ has infinitely many solutions, then $\alpha +\beta$ is equal to :

Imagine you are standing in a vast, three-dimensional space. You have three sheets of paper, each representing a linear equation: $x+y+z=6$ $2x+5y+\alpha z=\beta$ $x+2y+3z=14$

In a typical scenario, these three planes would meet at a single, unique point. However, we are looking for the condition where these planes intersect along an entire line, which corresponds to the case of infinitely many solutions.

To find this, we turn to the elegance of Cramer's Rule. We define the main determinant $D$ as a measure of the independence of these planes. If $D eq 0$, the planes are independent and intersect at a unique point.

If $D=0$, the planes are linearly dependent, allowing for the possibility of infinite intersections. We construct our determinant using the coefficients of our variables:

Expanding this along the first row, we calculate:

Simplifying this expression, we get:

For the system to have infinite solutions, we must have $D=0$, which immediately gives us $\alpha =8$.

Now that we have determined $\alpha$, we turn our attention to $\beta$. For the system to be consistent with infinite solutions, the sub-determinants must also vanish. We construct $D_1$ by replacing the first column of our coefficient matrix with the constants from the right-hand side:

Expanding this determinant, we get:

Breaking this down further:

This simplifies to:

Setting $D_1=0$ leads us directly to $\beta =36$.

We have navigated the geometry and the algebra to find $\alpha =8$ and $\beta =36$. The question asks for the sum $\alpha +\beta$.

Adding these together:

By understanding the geometric requirement of the planes and applying the algebraic rigor of determinants, we have solved the system. The final result is $44$.