Animated Solution for Mathematics - Matrices and Determinants: If the system of equations $\alpha x+y+z=5$, $x+2y+3z=4$, $x+3y+5z=\beta$, has infinitely many solutions, then the ordered pair $(\alpha ,\beta )$ is equal to :

A Journey into Linear Systems

Imagine you are standing in a vast, empty room with three sheets of glass—three planes—suspended in the air. In most cases, these planes intersect at a single point, like the corner of a room where two walls meet the ceiling.

However, we are looking for a more elusive configuration: where these three planes intersect not at a point, but along an entire line. Think of the pages of an open book meeting at the spine.

This is the geometric reality of "infinitely many solutions." It is a state of perfect, harmonious dependency that we can unlock using the power of linear algebra.

Cramer's Rule

To solve this, we turn to Cramer's Rule, the scalpel of linear algebra. For a system of equations, we define the main determinant, $\Delta$, using the coefficients of our variables $x$, $y$, and $z$.

If $\Delta eq 0$, the system is independent and has a unique solution. We seek the dependent case where the planes "collapse" into a line, which requires $\Delta =0$.

$\Delta =0$ is a necessary condition, but not a sufficient one. We must also ensure that the auxiliary determinants—${\Delta }_x$, ${\Delta }_y$, and ${\Delta }_z$—are zero to ensure the system is consistent. If $\Delta =0$ but $\Delta_x eq 0$, the planes are parallel and never meet, leading to no solution.

The Hunt for $\alpha$

Let us construct our main determinant, $\Delta$, from the coefficients of our system: $\alpha x+y+z=5$, $x+2y+3z=4$, and $x+3y+5z=\beta$. The matrix is defined as:

Expanding this along the first row, we perform the arithmetic:

Simplifying the terms inside the brackets:

For the system to have infinite solutions, we set $\Delta =0$. Thus, $\alpha -1=0$, which gives us $\alpha =1$.

The Hunt for $\beta$

Now that we know $\alpha =1$, we must find $\beta$. For the system to be consistent, ${\Delta }_x$ must also be zero. We form ${\Delta }_x$ by replacing the first column of our coefficient matrix with the constants $5$, $4$, and $\beta$:

Expanding this determinant:

Simplifying the expression:

Since we require ${\Delta }_x=0$ for consistency, we set $\beta -3=0$, which yields $\beta =3$.

We have navigated the geometry of three-dimensional space using the logic of determinants. We found that when $\alpha =1$ and $\beta =3$, the planes align perfectly, intersecting along a common line.

The final ordered pair is $(\alpha ,\beta )=(1,3)$.

Mathematics is not just about numbers; it is about understanding the structure of the universe. Whether you are solving for $\alpha$ or $\beta$, you are uncovering the hidden relationships between objects in space.