Chapter 9.2, Problem 41WE

To determine

To find: The solution of the given equations by using substitution method.

Answer to Problem 41WE

The solution of the given equations by using substitution method is $x=\frac{5}{3},y=\frac{2}{3}\text{ and }z=-\frac{2}{3}$

Explanation of Solution

Given information:

The given equations are,

  $\begin{array}{l}x+y+2z=1\\ x-y=1\\ x-z=2\end{array}$

The given equation can be written as:

  $\begin{array}{l}x+y+2z=1\\ 2z=1-x-y\\ z=\frac{1-x-y}{2}\end{array}$

Substitute $\frac{1-x-y}{2}$ for $z$

  $\begin{array}{l}x-z=2\\ x-\frac{1-x-y}{2}=2\\ 2x-1+x+y=4\\ 3x+y=5\end{array}$

Now solve the equations:

  $\begin{array}{l}x-y=1\\ 3x+y=5\end{array}$

The value of $x$ can be written as:

  $\begin{array}{l}x-y=1\\ x=1+y\end{array}$

Substitute $1+y$ for $x$

  $\begin{array}{l}3x+y=5\\ 3\left(1+y\right)=5\\ 3+3y=5\\ y=\frac{2}{3}\end{array}$

The value of $x$ is calculated as,

  $\begin{array}{l}x=1+y\\ x=1+\frac{2}{3}\\ x=\frac{5}{3}\end{array}$

The value of $z$ is calculated as,

  $\begin{array}{c}z=\frac{1-x-y}{2}\\ =\frac{1-\frac{5}{3}-\frac{2}{3}}{2}\\ =\frac{3-5-2}{6}\\ =-\frac{2}{3}\end{array}$

Thus, the solution of the given equations by using substitution method is $x=\frac{5}{3},y=\frac{2}{3}\text{ and }z=-\frac{2}{3}$