Linear algebra question...

• Sep 30th 2007, 07:34 PM
pakman
Linear algebra question...
Given a system of the form

-m1x1 + x2 = b1
-m2x2 + x2 = b2

where m1, m2, b1, and b2 are constants:

(a) Show that the system will have a unique solution if m1 != m2
(b) If m1 = m2, show that the system will be consistent only if b1 = b2
(c) Give a geometric interpretation to parts (a) and (b)

Not sure where to even begin with this, thanks in advance!
• Sep 30th 2007, 10:25 PM
earboth
Quote:

Originally Posted by pakman
Given a system of the form

-m1x1 + x2 = b1
-m2x2 + x2 = b2

where m1, m2, b1, and b2 are constants:

(a) Show that the system will have a unique solution if m1 != m2
(b) If m1 = m2, show that the system will be consistent only if b1 = b2
(c) Give a geometric interpretation to parts (a) and (b)

Not sure where to even begin with this, thanks in advance!

Hello,

first I assume that there is a typo:

$\displaystyle \begin{array}{l}-m_1 x_1 + x_2 = b_1 \\-m_2 x_1 + x_2 = b_2\end{array}$ . Subtract columnwise(?):

$\displaystyle -m_1x_1 + m_2 x_1 = b_1 - b_2~\iff~x_1 = \frac{b_1-b_2}{m_2-m_1}$ . As you easily can see there exists only a solution for $\displaystyle x_1$ if $\displaystyle m_1 \ne m_2$

If $\displaystyle m_1 = m_2$ then the 2 lines are parallel and there doesn't exist a common point, that means there doesn't exist a solution for $\displaystyle x_1$

If $\displaystyle m_1 = m_2~\wedge~b_1 = b_2$ the lines are equal that means there are infinite many common points.