Animated Solution for Mathematics - Matrices and Determinants: If the system of linear equations $2x+y-z=7$, $x-3y+2z=1$, $x+4y+\delta z=k$, where $\delta ,k\in \mathbb{R}$ has infinitely many solutions, then $\delta +k$ is equal to:

Imagine you are standing in a vast, three-dimensional space. You have three planes, each defined by a linear equation.

Usually, these planes might intersect at a single point, like the corner of a room. But today, we are looking for something more elegant. We are looking for a system that has infinitely many solutions.

Geometrically, this means the three planes do not meet at a single point; instead, they all intersect along a single, common line. Every point on that line is a solution. This is the core reality we must visualize.

Cramer's Rule

To translate this geometric intuition into algebra, we turn to Cramer's Rule. For a system of three linear equations in three variables to have infinitely many solutions, the main determinant of the coefficient matrix, denoted as $\Delta$, must be zero.

But that is not enough. If $\Delta =0$ but the system is inconsistent, we would have no solutions.

To guarantee consistency and infinite solutions, all auxiliary determinants—${\Delta }_x$, ${\Delta }_y$, and ${\Delta }_z$—must also be zero. This is a fundamental condition you must keep locked in your memory for JEE Advanced.

Solving for $\delta$

Let us construct our main determinant $\Delta$ from the coefficients of our system: $2x+y-z=7$, $x-3y+2z=1$, and $x+4y+\delta z=k$. The determinant is:

Expanding this along the first row, we get:

Simplifying this step-by-step:

For infinite solutions, we set $\Delta =0$:

We have found our first piece of the puzzle.

Finding $k$

Now, we need to find $k$. We use the auxiliary determinant ${\Delta }_z$ (or ${\Delta }_3$), where we replace the third column (the $z$-coefficients) with the constants $(7,1,k)$:

Expanding this along the first row:

Setting ${\Delta }_3=0$ for consistency:

We have successfully navigated the algebra. We found $\delta =-3$ and $k=6$. The question asks for the sum $\delta +k$.

It is a beautiful, clean result. By understanding the geometric condition of the planes and applying the algebraic constraints of Cramer's Rule, we have unlocked the solution.

Remember, in JEE, it is not just about the calculation; it is about understanding the structure of the problem. The final answer is 3.