Animated Solution for Mathematics - Matrices and Determinants: The equation $x+2y+2z=1$ and $2x+4y+4z=9$ have

Welcome, future engineer. Today we are not just solving a system of equations; we are embarking on a journey into the heart of three-dimensional geometry.

When you look at the system $x+2y+2z=1$ and $2x+4y+4z=9$, I want you to stop seeing variables and start seeing reality. Imagine you are standing in a vast, empty room where each equation represents a flat, infinite sheet of paper floating in space. Our mission is to determine how these two sheets interact.

In the world of 3D coordinate geometry, a linear equation of the form $ax+by+cz=d$ defines a plane. To describe the "tilt" or orientation of this plane, we use a normal vector, $\vec{n}=⟨a,b,c⟩$.

Think of this vector as a flagpole sticking straight out of the surface of the plane. If you know the direction of the flagpole, you know exactly how the plane is tilted.

For our first equation, $x+2y+2z=1$, the coefficients are $1,2,2$. Thus, our first normal vector is:

Now, look at the second equation: $2x+4y+4z=9$. Its normal vector is:

Here is where the magic happens. Look closely at ${\vec{n}}_1$ and ${\vec{n}}_2$. If you take ${\vec{n}}_1$ and multiply every component by $2$, you get ${\vec{n}}_2$.

Mathematically, we write this as:

This is a profound discovery! It means the "steering columns" of both planes are pointing in the exact same direction. If the planes are tilted in the exact same way, they must be parallel.

Parallel planes have two possibilities: they could be coincident (the same plane) or they could be distinct (separated by a gap). We must check the constants to distinguish between these cases.

Let's take our first equation, $x+2y+2z=1$, and multiply the entire expression by $2$:

Now, compare this to our second equation: $2x+4y+4z=9$. The left-hand sides are identical, which confirms they are parallel.

However, the right-hand sides are $2$ and $9$. Since $2 eq 9$, it is physically impossible for a point $(x,y,z)$ to satisfy both equations simultaneously.

The planes are distinct, parallel, and separated by a constant distance. They will never intersect. Therefore, there are zero solutions.

Since "zero" is not an option, we confidently select "None of these". This is the beauty of mathematics: it provides a rigorous, undeniable truth about the world. You have successfully navigated the trap.