## Linear Algebra and Its Applications, Exercise 2.2.8

Exercise 2.2.8. Consider the following system of linear equations:

$\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcl}u&+&v&+&2w&=&2 \\ 2u&+&3v&-&w&=&5 \\ 3u&+&4v&+&w&=&c \end{array}$

For what value of $c$ does this system have a solution?

Answer: We use elimination to attempt to solve the system, starting by multiplying the first equation by 2 and subtracting it from the second, and multiplying the first equation by 3 and subtracting it from the third:

$\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcl}u&+&v&+&2w&=&2 \\ 2u&+&3v&-&w&=&5 \\ 3u&+&4v&+&w&=&c \end{array} \Rightarrow \begin{array}{rcrcrcl}u&+&v&+&2w&=&2 \\ &&v&-&5w&=&1 \\ &&v&-&5w&=&c-6 \end{array}$

and then substract 1 times the second equation from the third:

$\begin{array}{rcrcrcl}u&+&v&+&2w&=&2 \\ &&v&-&5w&=&1 \\ &&v&-&5w&=&c-6 \end{array} \Rightarrow \begin{array}{rcrcrcl}u&+&v&+&2w&=&2 \\ &&v&-&5w&=&1 \\ &&&&0&=&c-7 \end{array}$

In order for this system to have a solution we must have $c - 7 = 0$ or $c = 7$.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.

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