Question

The general equation of a plane is Ax + By + Cz = D, where A, B, C, and D are real numbers and A is nonnegative. Find the equation of the plane containing

the points (2, 0, 0), (0, 6, 0), and (0, 0, 5). Show each step of your process. Then graph the plane.

Answer 1

You'll want to think of this as a problem with 3 equations and 3 unknowns.
First, rewrite the initial equation by dividing everything by D:

(A/D)x + (B/D)y + (C/D)z = 1

To make this easier, you can substitute in new variables: A/D = m, B/D = n, and C/D = o. So:

mx + ny + oz = 1

Now, you should put each point into this equation, which will give you 3 new equations:

2m + 0 + 0 = 1
0 + 6n + 0 =1
0 + 0 + 5o = 1

In this case you can solve each directly, otherwise you could use substitution:

m = 1/2, n = 1/6, o = 1/5

Now, put these values in the cleaned up equation:

1/2 x + 1/6 y + 1/5 z = 1

This is in standard form, however you can multiply each term by 30 to remove fractions:

15x + 5y + 6z = 30.