# Linear Algebra System Basic Proof

• January 18th 2010, 04:43 PM
WTFsandwich
Linear Algebra System Basic Proof
Given a system of the form

mx + y = c
nx + y = d

where c, d, m and n are constants:

A. Show that the system will have a unique solution if m does not equal n.
B. If m = n, show the system will only be consistent if c = d.
C. Give a geometric interpretation of parts A and B

I have no idea where to begin with that. I tried using elementary row operations on the augmented matrix, but I couldn't gather anything from it and I fear I may have done it incorrectly.

Any help would be greatly appreciated.
• January 18th 2010, 06:12 PM
Prove It
Quote:

Originally Posted by WTFsandwich
Given a system of the form

mx + y = c
nx + y = d

where c, d, m and n are constants:

A. Show that the system will have a unique solution if m does not equal n.
B. If m = n, show the system will only be consistent if c = d.
C. Give a geometric interpretation of parts A and B

I have no idea where to begin with that. I tried using elementary row operations on the augmented matrix, but I couldn't gather anything from it and I fear I may have done it incorrectly.

Any help would be greatly appreciated.

A. These are linear functions.

You should know that you can express a linear function in the form $y = ax + b$, with $a$ as the gradient and $b$ as the $y$ intercept.

Rearranging the equations gives:

$y = -mx + c$
$y = - nx + d$.

If $m = n$ then the gradients will be the same and the lines will be parallel. Therefore there will not be a solution.

For $m \neq n$ we have

$-mx + c = -nx + d$

$c - d = (m - n)x$

$x = \frac{c - d}{m - n}$.

Substituting back into one or both of the equations, you will see there is only one solution.

B. If $m = n$, like I said, the lines are parallel.

However if $c = d$ as well, the equations are in fact, identically equal. In other words, they are the same line.

C. Done in A and B - not sure if it's what you wanted...