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 independent, 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)

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


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