# Thread: Using Homogeneous Equations to prove the problems ?

1. ## Using Homogeneous Equations to prove the problems ?

Hi everyone,

This time I have a little bit confuse when I try to solve this question that I think it is obviously shown:

(a) Show that there is a line through any pair of points in the plane. [Hint: Every line has equation ax + by + c = 0, where a, b, c are not all zero].

(b) Generalize and show that there is a plane ax + by + cz + d = 0 through any three points in space.

(Source: Question 6, Section 1.3, Chapter 1, Linear Algebra with Applications 5th Edition, W. Keith Nicholson)

At first I think it is so easy but after that I can not think how can I prove when it is too obviously. It made me going crazy and frustrated. Or maybe I have some problem in thinking (stupid or something ?) that I can not solve this.

Thank you so much.

2. When something is "too obvious", do the obvious! Let the two points be $\displaystyle (x_0, y_0)$ and $\displaystyle (x_1, y_1)$. If that line has equation ax+ by+ c= 0, then the equation must satisfy $\displaystyle ax_0+ by_0+ c= 0$ and $\displaystyle ax_1+ by_1+ c= 0$.

Those are the same as $\displaystyle ax_0+ by_0= -c$ and $\displaystyle ax_1+ by_1= -c$. As long as you can solve those equations for a and b, the line exists (and is unique). Under what conditions can you not solve those equations? Treat that case separately.

3. Thank you for you very logical and smart way.

At first I think I must solve those equations but after that I asked myself "so what next ".

Now I have an answer. Thank you so much!