Animated Solution for Mathematics - Matrices and Determinants: If the system of equations $x+y+z=5$, $x+2y+3z=9$, $x+3y+\alpha z=\beta$ has infinitely many solutions, then $\beta -\alpha$ equals:

Welcome, future engineer! Today, we are not just solving a system of equations; we are exploring the architecture of 3D space. Imagine three planes, $P_1,P_2,$ and $P_3$, floating in the vastness of coordinate geometry.

Usually, these planes might meet at a single point, or perhaps not at all. But here, the problem whispers a secret: they intersect along an entire line. This is the condition for infinitely many solutions.

To unlock this, we turn to the powerful machinery of Cramer's Rule.

We start by examining the coefficient matrix. The determinant $D$ must vanish for the system to have infinitely many solutions:

If $D eq 0$, the system would have a unique solution. Since we are promised an infinite number of solutions, the planes must be linearly dependent.

Instead of blindly expanding, let's be elegant. We apply row operations: $R_2\to R_2-R_1$ and $R_3\to R_3-R_1$. This transforms our determinant into:

Expanding along the first column is now a breeze. We get:

Setting this to zero, we find $\alpha =5$. This is our first victory!

Now that we have $\alpha =5$, we must determine $\beta$. This is where the auxiliary determinant $D_x$ comes into play.

We construct $D_x$ by replacing the first column of our original matrix with the constants $(5,9,\beta )$:

We expand this along the first row, being careful with the signs:

Simplifying this step-by-step:

For the system to be consistent, we set $D_x=0$, which gives us $\beta =13$.

We have found the keys to the kingdom: $\alpha =5$ and $\beta =13$. The question asks for the value of $\beta -\alpha$.

And there it is! The final answer is 8.

It is elegant, precise, and deeply satisfying. You have successfully navigated the intersection of three planes and emerged with the correct parameters. Remember, in JEE Advanced, it is not just about the calculation; it is about understanding the geometric soul of the algebra.