Chapter 9.9, Problem 15WE

To determine

To calculate: To solve the equation and find values for u,v,w

Answer to Problem 15WE

  $\{u,v,w\}=\left\{-1,2,3\right\}$

Explanation of Solution

Given:

  $\begin{array}{l}3u+2v+w=4\\ 5u+3v-w=-2\\ 2u+w=1\end{array}$

Calculation:

Given equation are as follows:

  $\begin{array}{l}3u+2v+w=\mathrm{4.....................}(1)\\ 5u+3v-w=-\mathrm{2.....................}(2)\\ 2u+w=\mathrm{1.....................}(3)\end{array}$

Solve for v in equation 2:

  $\begin{array}{l}5u+3v-w=-2\\ 3v=-5u+w-2\\ v=\frac{-5u+w-2}{3}\mathrm{.....................}(4)\end{array}$

Put the value v from equation (4) in equation (1):

  $\begin{array}{l}3u+2v+w=4\\ 3u+2\left(\frac{-5u+w-2}{3}\right)+w=4\\ 9u+(-10u+2w-4)+3w=12\\ 9u-10u+2w-4+3w=12\\ -u+5w=16\\ u=5w-\mathrm{16.....................}(5)\end{array}$

Put the value u from equation (5) in equation (3):

  $\begin{array}{l}2u+w=1\\ 2(5w-16)+w=1\\ 10w-32+w=1\\ 11w=33\\ w=\mathrm{3.....................}(6)\end{array}$

Put the value of w from equation (6) in equation (5) and calculate u:

  $\begin{array}{l}u=5w-16\\ u=5\left(3\right)-16\\ u=-\mathrm{1.....................}(7)\end{array}$

Put the value of w and u from equation (6) and (7) in equation (4) and calculate v:

  $\begin{array}{l}v=\frac{-5u+w-2}{3}\\ v=\frac{-5\left(-1\right)+\left(3\right)-2}{3}\\ v=\frac{5+3-2}{3}\\ v=\frac{6}{3}\\ v=\mathrm{2.....................}(8)\end{array}$

Hence, from equation (6),(7),(8): $\{u,v,w\}=\left\{-1,2,3\right\}$

Conclusion:

  $\{u,v,w\}=\left\{-1,2,3\right\}$