Chapter 9.9, Problem 23WE

To determine

To find: whether there is any solution and also find the general expression for the solution.

Answer to Problem 23WE

The general expressions for the solutions of the given system is $\{(2z,5z,z)\}.$

Explanation of Solution

Given:

  $x-2z=0$

  $2x-y+z=0$

  $x-y+3z=0$

Calculation:

Let us first transform the given system of equations into a triangular system.

Multiply first equation by $-2$ and add the result to the second equation.

  $\begin{array}{l}2x-y+z=0\\ -2(x-2z=0)\\ -y+5z=0\end{array}$$\begin{array}{l}x-2z=0\\ -y+5z=0\\ x-y+3z=0\end{array}$

Then, multiply the first equation by $-1$ and add the result to the third equation.

  $\begin{array}{l}-1(x-2z=0)\\ x-y+3z=0\\ -y+5z=0\end{array}$$\begin{array}{l}Resultingsystem\\ x-2z=0\\ -y+5z=0\\ -y+5z=0\end{array}$  $(1)$

In $(1)$ , subtract the third equation from the second equation.

  $\begin{array}{l}-y+5z=0\\ -y+5z=0\\ 0=0\end{array}$$\begin{array}{l}Resultingsystem\\ x-2z=0\\ -y+5z=0\\ 0=0\end{array}$  $(2)$

The third equation in the resulting system shows that the given system is dependent and has infinitely many solutions. The solutions for $x$ and $y$ can be determined in terms of $z$ .

In $(2)$, the value of $x$ can be obtained by solving the first equation.

  $\begin{array}{l}x-2z=0\\ x=2z\end{array}$

The value of $y$ can be obtained by solving the second equation.

  $\begin{array}{l}-y+5z=0\\ y=5z\end{array}$

Therefore, the general expressions for the solutions of the given system is $\{(2z,5z,z)\}.$

Conclusion:

The general expressions for the solutions of the given system is $\{(2z,5z,z)\}.$