Geometrical interpretation of solving a $3 \times 3$ system of equations

Question

Solve the following system of equations and give a geometrical interpretation of the result. \begin{align*} x + y + z &= 6\\ 2x + y − 3z &= -5\\ 4x − 5y + z &= −3 \end{align*}

I know that each equation represents a plane in $3$-$D$ and that $x,y,z = 1,2,3.$ I want to show how i got that answer but I am have trouble formatting it. I used row reduction. But what do I do with that information. What is geometrical interpretation?

Answer 1

It is the single point of concurrence between 3 planes.

If one variable is eliminated from the three, say $z$, then two lines intersect giving $x,y$ intersection.

Answer 2

The catch is the fact that you are able to use row reduction to get an answer just because the system of three equations form a consistent system. In a geometrical sense, just because three lines (each formed by the intersection of 2 planes) are concurrent, you are able to get a valid solution (a point in 3D )by row reduction.(Gaussian Elimination I guess.)