## Linear Algebra and Its Applications, Review Exercise 1.19

Review exercise 1.19. Solve the following systems of equations using elimination and back substitution: $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcr}u&+&v&+&w&=&0 \\ u&+&2v&+&3w&=&0 \\ 3u&+&5v&+&7w&=&1 \end{array}$    and $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcr} u&+&v&+&w&=&0 \\ u&+&v&+&3w&=&0 \\ 3u&+&5v&+&7w&=&1 \end{array}$ $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcr}u&+&v&+&w&=&0 \\ u&+&2v&+&3w&=&0 \\ 3u&+&5v&+&7w&=&1 \end{array}$

and subtract 1 times the first equation from the second and 3 times the first equation from the third to obtain the following system: $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcr}u&+&v&+&w&=&0 \\ &&v&+&2w&=&0 \\ &&2v&+&4w&=&1 \end{array}$

We can then subtract 2 times the second equation from the third to produce the following system: $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcr}u&+&v&+&w&=&0 \\ &&v&+&2w&=&0 \\ &&&&0&=&1 \end{array}$

We have reached a contradiction, so this system has no solution.

We next look at the system $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcr} u&+&v&+&w&=&0 \\ u&+&v&+&3w&=&0 \\ 3u&+&5v&+&7w&=&1 \end{array}$

We subtract 1 times the first equation from the second and 3 times the first equation from the third to obtain the following system: $\setlength\arraycolsep{0.2em}\begin{array}{rcrcrcr} u&+&v&+&w&=&0 \\ &&&&2w&=&0 \\ &&2v&+&4w&=&1 \end{array}$

From the second equation we have $w = 0$. Substituting $w$ into the third equation yields $v = \frac{1}{2}$. Substituting $v$ and $w$ into the first equation yields $u = -v - w = -\frac{1}{2}$. The solution is therefore $\setlength\arraycolsep{0.2em}\begin{array}{rcr}u&=&-\frac{1}{2} \\ v&=&\frac{1}{2} \\ w&=&0 \end{array}$

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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