What Value of k Ensures Consistency in These Simultaneous Equations?

[Gregg]

Homework Statement

Show that the simultanbeous equations

6x-7y+2z=4

6x-y-z=7

2x-3y+z=k

where k is a constant, are consisten only when k=1.

The Attempt at a Solution

Don't know how to start, the determinent of the co-efficient matrix is -9. This means they are independant, which means I can't express multiples of (1) and (2) for (3) right? I tried to get x and y in terms of z, then substitute for (3)... Doesn't work. I need the method.

 

[statdad]

Set up the augmented matrix of your system of equations (since the order in which the equations are given is unimportant, you can set things up this way)

<br /> \begin{pmatrix}<br /> 2 &amp; -3 &amp; \hphantom{-}1 &amp; k \\<br /> 6 &amp; -7 &amp; \hphantom{-}2 &amp; 4\\<br /> 6 &amp; -1 &amp; -1 &amp; 7<br /> \end{pmatrix}<br />


Use row operations to reduce this to row reduced echelon form. You'll see why when you reach that form.