Animated Solution for Mathematics - Matrices and Determinants: If the system of equations $x+2y-3z=2$ $2x+\lambda y+5z=5$ $14x+3y+\mu z=33$ has infinitely many solutions, then $\lambda +\mu$ is equal to:

Welcome, future engineers and physicists. Today, we are not just solving a system of linear equations; we are exploring the architecture of three-dimensional space.

Imagine you are standing in a room where three massive, flat planes intersect. Usually, three planes meet at a single point, like the corner of a room. But what if they meet along a line, like the spine of a book?

When a system of equations has infinitely many solutions, it means our three planes are locked in a perfect, synchronized dance, sharing an entire line of points. This is the geometric reality behind the algebraic condition $D=D_x=D_y=D_z=0$.

Why Cramer's Rule?

We are given the system:

We need to find $\lambda$ and $\mu$. Many students would immediately jump into Gaussian elimination, but let's be smarter. We have a powerful tool: Cramer's Rule.

The condition for infinitely many solutions is that the main determinant $D$ and all numerator determinants $D_x,D_y,D_z$ must be zero. Here is the secret: look at the $z$-column. It contains $\mu$.

If we calculate $D_z$ by replacing the $z$-column with the constants $(2,5,33)$, $\mu$ vanishes! This is our strategic opening.

Let's construct $D_z$:

Expanding along the first row, we get:

Simplifying this, we find $33\lambda -15+8+12-28\lambda =0$, which elegantly reduces to $5\lambda +5=0$. Thus, $\lambda =-1$.

With $\lambda$ in hand, the system becomes much clearer. Now, we turn our attention to the main determinant $D$ to find $\mu$:

Expanding this, we get:

This simplifies to $-\mu -15-4\mu +140-60=0$, or $-5\mu +65=0$. Therefore, $\mu =13$.

The question asks for $\lambda +\mu$. With $\lambda =-1$ and $\mu =13$, the sum is simply $12$.

We have navigated the geometry of intersecting planes and used the precision of determinants to find our answer. Remember, in JEE Advanced, the math is not just about calculation; it is about visualization and strategy.

You have successfully mastered the spine of the book. Keep this intuition, and no system of equations will ever intimidate you again.