Why can we set two variables zero to find a point on a plane?

[Mathematicsss]

Homework Statement

Why is that we can set two variables zero in an equation of a plane to find a point on that plane? What is the proof for this?

Homework Equations

The Attempt at a Solution

 

[Charles Link]

The equation of a plane is of the form $Ax+By +Cz+D=0$. The plane will normally cross the z-axis, so that we can set x=0 and y=0, and compute the z where it crosses the z-axis. (Anywhere along the z-axis, both x and y are zero). If $C=0$, and $D \neq 0$, it doesn't cross the z-axis. Similarly for the other axes.

 

[Mark44]

If you have one equation in one variable, such as 2x - 3 = 5, the equation has a single solution. Geometrically, you're looking for a value of x (a number on the x-axis) that makes the equation a true statement.
If you have two equations in two variables, this represents two lines in the plane that might or might not intersect.
If you have one equation in two variables, the equation represents a line, meaning that the system (of one equation) has an infinite number of solutions -- any point on the line. You can set either variable to whatever value you like, and from this, can determine the other variable at that point.
The situation is similar if you have one equation in three variables. You can set any two of the variables to arbitrary values, which will uniquely determine the value of the third variable.