Animated Solution for Mathematics - Matrices and Determinants: If the system of equations $$\begin{gathered} x-2y+3z=9 \\ 2x+y+z=b \\ x-7y+az=24 \end{gathered}$$, has infinitely many solutions, then $a-b$ is equal to

Imagine you are standing in a vast, three-dimensional space. You have three sheets of paper, each representing a linear equation: $x-2y+3z=9$, $2x+y+z=b$, and $x-7y+az=24$.

Usually, these planes intersect at a single point, like the corner of a room. However, this system has infinitely many solutions.

Geometrically, this means these three planes are not meeting at a single point; instead, they are all meeting along a single, common line. Think of the pages of an open book meeting at the spine. That spine is our line of infinite solutions.

Our first task is to find the value of $a$. We look at the coefficient matrix of our system:

For the system to have infinitely many solutions, the determinant of this matrix, $\Delta$, must be zero. Let us set it up:

Now, we expand this determinant along the first row:

Simplifying this, we get:

Solving for $a$, we find $a=8$. We have successfully unlocked the first mystery!

Now that we know $a=8$, we need to find $b$. We return to our condition for infinite solutions: the auxiliary determinants must also be zero.

We focus on ${\Delta }_x$, where we replace the first column of our coefficient matrix with the constant terms from the right side of our equations: $9$, $b$, and $24$.

Let us expand this carefully along the first row:

This simplifies to:

Combining the terms, we have $15-5b=0$. Thus, $5b=15$, and $b=3$.

We have found $a=8$ and $b=3$. The problem asks us for the value of $a-b$.

Substituting our hard-earned values, we get:

It is truly elegant how the abstract condition of infinite solutions guides us through the matrix algebra to a single, concrete number. The final result is $5$.