# Finding value of $c$ [duplicate]

Possible Duplicate:
Solution of 3 equations in 3 unknowns

I need help finding the value of $c$ which makes it possible to solve: $$\begin{array}{lcl} u + v + 2w & = & 2 \\ 2u + 3v - w & = & 5 \\ 3u + 4v + w & = & c \end{array}$$

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 This is an exact duplicate, but I happened to see this one first, hence the answer below. – Brian M. Scott Oct 9 '12 at 9:45

## marked as duplicate by Brian M. Scott, draks ..., Matt Pressland, robjohn♦Oct 9 '12 at 10:35

$$\left[\begin{array}{rrr|r} 1&1&2&2\\ 2&3&-1&5\\ 3&4&1&c \end{array}\right]\;.$$
$$\left[\begin{array}{rrr|c} 1&1&2&2\\ 0&1&-5&1\\ 0&1&-5&c-6 \end{array}\right]\;.\tag{1}$$
Now you can either stop and think about the equations corresponding to the bottom two rows of $(1)$ (what does $c$ have to be in order for them to be consistent?), or finish the row-reduction and then think about what $c$ has to be to avoid having an inconsistent system.