Animated Solution for Mathematics - Matrices and Determinants: The following system of linear equations $7x+6y-2z=0$, $3x+4y+2z=0$, $x-2y-6z=0$, has

Imagine you are standing in a three-dimensional space, looking at three distinct planes. Each equation in our system, $7x+6y-2z=0$, $3x+4y+2z=0$, and $x-2y-6z=0$, represents a flat surface slicing through the origin $(0,0,0)$.

Because every equation is set to zero, we know for a fact that the origin is a meeting point for all three. This is the hallmark of a homogeneous system.

The core mystery is whether these planes meet only at this single point, or if they share an entire line of points.

The Gatekeeper of Solutions

Before we get lost in the algebra, we must consult the gatekeeper: the determinant $\Delta$. The determinant tells us if the system is 'full rank' or if there is a hidden redundancy. We construct our matrix of coefficients:

Expanding this along the first row, we perform the calculation:

As we simplify the terms inside the brackets, we see the numbers begin to align in a way that feels almost poetic:

When $\Delta =0$, the system is singular. It tells us that one of these equations is actually a combination of the others. The planes are not independent; they are locked in a dance, intersecting along a common line.

We have moved beyond the trivial solution $(0,0,0)$ and into the realm of infinitely many solutions.

Now that we know there are infinitely many solutions, our goal is to find the 'rule' that governs them. We look at our first two equations:

Look at the $z$ terms. They are $-2z$ and $+2z$. This is a gift! By adding these two equations, we instantly eliminate $z$:

Dividing by $10$, we find the elegant relationship $x+y=0$, or $y=-x$. We have successfully reduced the complexity of the system.

Now, we substitute $y=-x$ back into our second equation to find the link to $z$:

This simplifies beautifully to $x=2z$.

We have arrived at the heart of the system. We found that for any value of $z$, $x$ must be $2z$, and consequently, $y$ must be $-2z$.

The entire set of solutions is defined by the line $(2z,-2z,z)$. This confirms that the system has infinitely many solutions, and specifically, those solutions satisfy the condition $x=2z$ and $y=-2z$.

Take a moment to appreciate the elegance of this result. We started with three complex equations and stripped away the noise to reveal a simple, linear relationship.

This is the power of linear algebra—turning chaos into a clear, geometric path. You have mastered the logic of the homogeneous system.