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

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Set up your augmented matrix in the usual way:

$$\left[\begin{array}{rrr|r} 1&1&2&2\\ 2&3&-1&5\\ 3&4&1&c \end{array}\right]\;.$$

Then row-reduce it; reducing the first column, for instance, yields

$$\left[\begin{array}{rrr|c} 1&1&2&2\\ 0&1&-5&1\\ 0&1&-5&c-6 \end{array}\right]\;.\tag{1}$$

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.

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If you are lucky in your row reduction, the first thing you try to do is add the first two equations and it comes out straight away! –  Ragib Zaman Oct 9 '12 at 9:43
    
@Ragib: If you do a standard row reduction, you’ll have no reason to perform that operation. You’re quite right that in this particular case there is a shortcut, but my answer is intended to demonstrate a systematic approach. –  Brian M. Scott Oct 9 '12 at 9:46
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