Animated Solution for Mathematics - Matrices and Determinants: If the system of equations $kx+y+2z=1,3x-y-2z=2,-2x-2y-4z=3$ has infinitely many solutions, then $k$ is equal to

Imagine you are standing in a three-dimensional space, looking at three distinct planes. In a typical system of linear equations, these planes intersect at a single point, giving you a unique solution.

But today, we are exploring a special case: the system has infinitely many solutions. This means these planes are not just meeting at a point; they are dancing together, intersecting along a common line or perhaps even coinciding entirely. This is the geometric soul of the problem we are solving today.

When we face a system of equations like $kx+y+2z=1$, $3x-y-2z=2$, and $-2x-2y-4z=3$, our first instinct should be to reach for our most reliable tool: Cramer's Rule.

We know that for a system to have infinitely many solutions, the main determinant $\Delta$ must be zero. But remember, this is only the first gate.

We must also ensure that the determinants ${\Delta }_x$, ${\Delta }_y$, and ${\Delta }_z$ are all zero. If $\Delta =0$ but one of the others is not, the planes are parallel and never meet, leaving us with no solution at all.

Let us construct our main determinant $\Delta$ using the coefficients of $x,y,z$:

As we expand this along the first row, we get:

Calculating the terms inside, we find:

Suddenly, the $k$ term vanishes because $4-4=0$. We are left with $0+16-16=0$. This confirms that $\Delta$ is zero regardless of $k$. The system is poised for infinite solutions, but we still need to find the specific $k$ that makes it happen.

Since $\Delta$ didn't help us find $k$, we pivot to the secondary condition: ${\Delta }_y=0$. We construct ${\Delta }_y$ by replacing the second column of our original matrix with the constants from the right-hand side:

Expanding this along the first row, we get:

Simplifying this, we have:

This reduces to ${\Delta }_y=-2k+16+26$, or simply ${\Delta }_y=-2k+42$.

Now, we set ${\Delta }_y=0$ to satisfy the condition for infinite solutions:

With a simple algebraic step, $2k=42$, we arrive at $k=21$.

It is a beautiful moment of clarity. We didn't just solve an equation; we navigated the constraints of 3D space to find the exact condition where these planes align perfectly. Keep this logic in your heart: $\Delta =0$ is the necessary condition, but the consistency of the system is the true test.