Animated Solution for Mathematics - Matrices and Determinants: Consider the following system of equations: $x+2y-3z=a,2x+6y-11z=b,x-2y+7z=c$ where $a,b$ and $c$ are real constants. Then the system of equations :

Imagine you are standing in a three-dimensional space, looking at three distinct planes. Each equation in our system, $x+2y-3z=a$, $2x+6y-11z=b$, and $x-2y+7z=c$, represents one of these planes.

The question of whether this system has a solution is essentially asking: do these three planes meet? Do they collide at a single point, share a line, or perhaps never meet at all?

The Determinant

To answer this, we turn to the most powerful tool in our linear algebra toolkit: the determinant of the coefficient matrix, $\Delta$. Think of $\Delta$ as the 'gatekeeper' of the system. It tells us whether the system is well-behaved.

We construct it by extracting the coefficients of $x$, $y$, and $z$:

Expanding this along the first row, we carefully compute:

Crunching the numbers, we get $1(42-22)-2(14+11)-3(-4-6)$, which simplifies to $20-50+30=0$. The gatekeeper has spoken: $\Delta =0$.

This tells us that the system is not 'well-behaved' in the sense of having a unique solution. We are now at a crossroads: the system either has infinite solutions or no solution at all.

To distinguish between these two paths, we use Cramer's Rule. For the system to have infinite solutions, the numerator determinants—${\Delta }_x$, ${\Delta }_y$, and ${\Delta }_z$—must also be zero.

Let's calculate ${\Delta }_x$ by replacing the first column of our original matrix with the constants $a$, $b$, and $c$:

Expanding this along the first row, we get:

Grouping the terms, we find ${\Delta }_x=20a-14b-22c+6b+18c$. Simplifying this expression, we obtain:

For the system to have infinite solutions, we require ${\Delta }_x=0$. Setting our expression to zero, we get $4(5a-2b-c)=0$.

This simplifies beautifully to the condition:

This is the condition that forces the planes to align in a way that they share an infinite number of points. It is a moment of mathematical elegance—where the chaos of three variables and three constants collapses into a single, clean, and powerful relationship. You have successfully navigated the system and found the key to its infinite nature!