Animated Solution for Mathematics - Matrices and Determinants: If the system of linear equations $3x+y+\beta z=3$, $2x+\alpha y-z=-3$, $x+2y+z=4$ has infinitely many solutions, then the value of $22\beta -9\alpha$ is :

Welcome, future engineer. Today, we are not just solving a system of equations; we are exploring the architecture of three-dimensional space. When you look at the system $3x+y+\beta z=3$, $2x+\alpha y-z=-3$, and $x+2y+z=4$, I want you to see three planes—three sheets of paper floating in space.

Usually, three planes intersect at a single point, like the corner of a room. But the problem tells us something fascinating: this system has infinitely many solutions. This means our three planes are meeting along a common line, locked in a dance of linear dependence.

To find the values of $\alpha$ and $\beta$ that allow this, we must use the language of determinants.

In the world of linear systems, the determinant of the coefficient matrix, which we denote as $\Delta$, acts as a gatekeeper. If $\Delta eq 0$, the system has a unique solution. Since the system has infinitely many solutions, we must have $\Delta =0$.

Let us construct our matrix:

Expanding this along the first row:

Simplifying this, we obtain:

This is our first bridge. It connects $\alpha$ and $\beta$, but we cannot cross it yet because we have two unknowns and only one equation.

Here is where the master educator's intuition comes in. If we calculate ${\Delta }_z$, we replace the third column (the $z$-coefficients) with the constants on the right side of the equations. Because the $z$-column contains $\beta$, replacing it effectively removes $\beta$ from the determinant entirely.

Let us set up ${\Delta }_z=0$:

Expanding this along the first row:

Now, watch the algebra unfold with precision:

We have found $\alpha$. Now, we return to Equation (1) to find $\beta$. Substitute $\alpha =-\frac{19}{9}$ into $3\alpha +4\beta -\alpha \beta +3=0$:

Combining the constants $-\frac{19}{3}+3$ gives us $-\frac{10}{3}$. Moving it to the other side:

We have our values: $\alpha =-\frac{19}{9}$ and $\beta =\frac{6}{11}$. The question asks for the value of $22\beta -9\alpha$.

Performing the final calculation:

The complexity collapses into a single, elegant integer: 31. You have navigated the geometry, mastered the determinant, and executed the algebra.