Animated Solution for Mathematics - Matrices and Determinants: The values of $\lambda$ and $\mu$ such that the system of equations $x+y+z=6$, $3x+5y+5z=26$, $x+2y+\lambda z=\mu$ has no solution, are :

Imagine you are standing in a vast, three-dimensional room. Each of the equations in our system, $x+y+z=6$, $3x+5y+5z=26$, and $x+2y+\lambda z=\mu$, represents a flat, infinite plane slicing through this space.

When we ask for a 'solution' to this system, we are asking for the point where all three planes meet. We are investigating the specific conditions under which these planes are arranged such that there is no single point common to all three.

The Power of Elimination

Solving for three variables simultaneously can feel like trying to untangle a knot in the dark. Consider the first two equations: $x+y+z=6$ and $3x+5y+5z=26$.

There is a hidden symmetry here. If we multiply the first equation by $5$, we obtain:

Now, compare this to the second equation. The $y$ and $z$ terms are identical. By subtracting the second equation from our modified first one, the $y$ and $z$ terms vanish, leaving us with:

We have successfully pierced the veil of this 3D system by finding the value of $x$ in seconds.

The 2D Reduction

With $x=2$ in our pocket, the problem transforms. We substitute this value back into our equations.

The first equation, $x+y+z=6$, becomes $2+y+z=6$, which simplifies to:

The third equation, $x+2y+\lambda z=\mu$, becomes $2+2y+\lambda z=\mu$, or:

We have successfully collapsed our 3D problem into a 2D problem: two lines on a flat $y-z$ plane. The question of 'no solution' for the 3D system is now equivalent to asking: when do these two lines never intersect?

The Parallelism Trap

For two lines to never intersect, they must be strictly parallel. If they were to cross at even a single point, we would have a unique solution. If they were to lie exactly on top of each other, we would have infinite solutions.

To ensure they are parallel, the ratio of their coefficients must be equal. Looking at our lines, the ratio of the $y$-coefficients is $1/2$, and the ratio of the $z$-coefficients is $1/\lambda$. Setting these equal:

This is the necessary condition for the lines to be parallel.

We must avoid the trap of coincidence. If the ratio of the constant terms also matches the coefficient ratio, the lines will overlap, creating infinite solutions.

We require the constant ratio to be different:

Solving this inequality, we get $\mu - 2 eq 8$, which means $\mu eq 10$. If $\mu$ were $10$, the lines would be identical.

By ensuring $\mu eq 10$, we guarantee that the lines are parallel and distinct, meaning they will never touch. The keys to this geometric impossibility are $\lambda =2$ and $\mu eq 10$.