Chapter 2.2, Problem 16E

To determine

To solve: the system of equations.

Answer to Problem 16E

  $\begin{array}{l}x=0.5\\ y=-0.75\\ z=0.2\end{array}$

Explanation of Solution

Given information:

  $\begin{array}{c}1.8x-z=0.7\mathrm{...}\text{(1)}\\ 1.2+z=-0.7\mathrm{...}\text{(2)}\\ 1.5x-3y=3\mathrm{...}\text{(3)}\end{array}$

Calculation:

Solve equation (3) for x as follows:

  $\begin{array}{c}1.5x-3y=3\\ 1.5x=3y+3\\ x=\frac{3}{1.5}y+\frac{3}{1.5}\\ x=2y+2\end{array}$

Solve equation (2) for y as follows:

  $\begin{array}{l}1.8x-z=0.7\\ 1.8\left(2y+2\right)-z=0.7\left(\text{Substitute }x\right)\\ 3.6y+3.6-z=0.7\\ 3.6y=-2.9+z\\ y=\frac{-2.9+z}{3.6}\mathrm{....}\left(4\right)\end{array}$

Solve equation (2) for y as follows:

  $\begin{array}{l}1.2y+z=-0.7\\ \mathrm{1..2}y=-0.7-z\\ y=\frac{-0.7-z}{1.2}\mathrm{....}\left(5\right)\end{array}$

Compare equation (4) and (5) and solve equation for z as follows:

  $\begin{array}{c}\frac{-2.9+z}{3.6}=\frac{-0.7-z}{1.2}\\ 1.2\left(-2.9+z\right)=3.6\left(-0.7-z\right)\left(\text{Cross-multiplication}\right)\\ -2.9+z=3\left(-0.7-z\right)\\ -2.9+z=-2.1-3z\\ 4z=0.8\\ z=\frac{0.8}{4}\\ =0.2\end{array}$

Put the value of z in equation (4) and solve as follows:

  $\begin{array}{c}y=\frac{-0.7-z}{1.2}\\ =\frac{-0.7-0.2}{1.2}\\ =-\frac{-0.9}{1.2}\\ =-0.75\end{array}$

Put the value of y in equation (3) and solve as follows:

  $\begin{array}{c}x=2\left(-0.75\right)+2\\ x=-1.5+2\\ =0.5\end{array}$

The values of x, y and z are $0.5,-0.75$ and $0.2$ respectively.