Animated Solution for Mathematics - Matrices and Determinants: For the system of equations $x+y+z=6$, $x+2y+az=10$, $x+3y+5z=\beta$, which one of the following is NOT true?

Imagine you are standing in a 3D coordinate system. You have three planes, each defined by an equation: $x+y+z=6$ $x+2y+az=10$ $x+3y+5z=\beta$

These are physical surfaces in space. Our goal is to determine how they interact: whether they meet at a single point, form a line of intersection, or never meet at all.

To understand the behavior of this system, we must first look at the coefficient matrix. We define the primary determinant, $\Delta$, as:

This determinant is the "gatekeeper." If $\Delta eq 0$, the system is well-behaved and possesses a unique solution. If $\Delta =0$, the system is singular, and we enter the territory of infinite solutions or no solutions.

Let us expand this determinant carefully along the first row:

Simplifying this, we get:

Combining the terms, we find $\Delta =10-3a-5+a+1$, which simplifies to:

For a unique solution, we require $\Delta eq 0$, which implies $6 - 2a eq 0$, or simply $a eq 3$.

What happens when $a=3$? This is the moment of truth. When $a=3$, our determinant $\Delta$ vanishes, becoming zero, and the system becomes singular.

We must now determine if it is inconsistent (no solution) or dependent (infinitely many solutions). We invoke Cramer's Rule and calculate ${\Delta }_x$, where we replace the first column of the coefficient matrix with the constants from the right-hand side:

Expanding this:

Simplifying further:

The constants are $6-50+30=-14$, and the variables are $3\beta -2\beta =\beta$. Thus:

If $\beta =14$, then ${\Delta }_x=0$. Since $\Delta =0$ and ${\Delta }_x=0$ (and similarly ${\Delta }_y={\Delta }_z=0$), the system has infinitely many solutions.

If $\beta eq 14$, then $\Delta_x eq 0$. Since $\Delta =0$ but $\Delta_x eq 0$, the system is inconsistent and has no solution.

Regarding the claim that the "System has a unique solution for $a = 3, \beta eq 14$": we have proven that when $a=3$, $\Delta$ is zero. A unique solution is mathematically impossible when the determinant is zero. Therefore, this statement is fundamentally false.