Animated Solution for Mathematics - Matrices and Determinants: If the system of equations $2x-y+z=4$ $5x+\lambda y+3z=12$ $100x-47y+\mu z=212$ has infinitely many solutions, then $\mu -2\lambda$ is equal to

Imagine you are standing in a vast, three-dimensional space. Before you, there are three flat, infinite sheets—planes—stretching out into the void.

Each of these planes is defined by a simple linear equation: $2x-y+z=4$, $5x+\lambda y+3z=12$, and $100x-47y+\mu z=212$.

In most cases, these three planes would meet at a single, solitary point in space. But today, the universe of this problem is different. We are told this system has infinitely many solutions.

Visually, this means these three planes are not just colliding at a point; they are all slicing through space to meet along a single, shared line. That line is the locus of all our solutions, a continuous path of points that satisfy all three equations simultaneously.

When we face a system like this, our goal is to find the parameters $\lambda$ and $\mu$ that force this geometric alignment. We turn to the elegant machinery of Cramer's Rule.

For a system to have infinitely many solutions, the main determinant $D$ must vanish, and the auxiliary determinants $D_x$, $D_y$, and $D_z$ must also be zero.

But here is where the master educator's intuition comes in: do not rush into the main determinant $D$! If you calculate $D$ first, you will find yourself staring at an equation with two variables, $\lambda$ and $\mu$, trapped together.

Instead, we look for the path of least resistance. We construct $D_z$ by replacing the $z$-coefficients with the constant terms from the right side of our equations.

Why? Because in this specific configuration, $D_z$ contains only $\lambda$. It is a tactical strike to isolate our first variable.

Let us construct $D_z$ carefully:

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

Simplifying this, we get:

As we distribute the terms, the complexity melts away:

Grouping the $\lambda$ terms gives us $24\lambda +48=0$, which leads us directly to $\lambda =-2$. The first piece of the puzzle is in our hands.

With $\lambda =-2$ secured, the path to $\mu$ is clear. We now set the main determinant $D$ to zero, substituting our known value of $\lambda$:

Expanding this again along the first row, we calculate:

This simplifies to:

Distributing the constants, we find:

Combining like terms, we arrive at $\mu -53=0$, or $\mu =53$. The geometry is satisfied, the algebra is complete, and the variables are revealed.

Finally, we calculate the requested expression:

We have navigated the system, avoided the traps, and arrived at the truth. This is the beauty of linear algebra—a perfect, logical symphony.