Animated Solution for Mathematics - Matrices and Determinants: The system of equations $x+y+z=6$ $x+2y+5z=9,$ $x+5y+\lambda z=\mu ,$ has no solution if

Imagine you are standing in a vast, three-dimensional space. You have three flat, infinite sheets of glass—these are your planes, defined by the equations $x+y+z=6$, $x+2y+5z=9$, and $x+5y+\lambda z=\mu$.

In a perfect world, these three planes would meet at a single, beautiful point, representing the unique solution $(x,y,z)$. But today, we are looking for the chaos—the moment when these planes refuse to meet. We are looking for the condition of no solution.

The Determinant $D$

To understand why a system fails, we must look at the coefficient matrix $D$. Think of $D$ as the 'DNA' of the system; it tells us whether the equations are linearly independent.

If $D eq 0$, the planes are oriented such that they must intersect at exactly one point. If $D=0$, the planes are 'linearly dependent'—they are either parallel or they intersect in a way that does not provide a single point of convergence.

We calculate the determinant:

Expanding this along the first row, we get:

As we simplify this, the tension builds. We find $D=2\lambda -25-\lambda +5+3$, which collapses beautifully into $D=\lambda -17$.

For the system to have any chance of being inconsistent, we must have $D=0$. Thus, the first piece of our puzzle falls into place: $\lambda =17$. If $\lambda$ were anything else, the system would have a unique solution, and our search for 'no solution' would be over before it began.

Distinguishing Inconsistency from Infinity

Now, here is where most students stumble. They find $\lambda =17$ and stop. But wait! If $D=0$, the system could either have no solution or infinitely many solutions.

To tell the difference, we look at the 'replacement' determinants, specifically $D_z$. Think of $D_z$ as the 'test of consistency'. We replace the third column of our matrix with the constants from the right-hand side of our equations:

Expanding this, we get:

Simplifying this expression, we arrive at $D_z=2\mu -45-\mu +9+18$, which simplifies to $D_z=\mu -18$.

If $D_z$ were zero, the planes would be perfectly aligned to intersect along a line, giving us infinitely many solutions. But we want no solution. We want the planes to be parallel or shifted such that they never touch.

Therefore, we demand that $D_z eq 0$. This leads us to the final condition: $\mu - 18 eq 0$, or simply $\mu eq 18$.

When you look at the final result—$\lambda =17$ and $\mu eq 18$—you aren't just looking at numbers. You are looking at the geometric configuration where the planes are forced into a state of perpetual disagreement.

You have mastered the logic of the system, navigated the trap of infinite solutions, and arrived at the truth. Every time you face a system of equations, remember: you are not just solving for $x,y,$ and $z$; you are orchestrating the geometry of space itself.