Chapter 10.2, Problem 9E

To determine

Using back substitution solve the triangular system.

Answer to Problem 9E

The values $x,y,z$using back substitution of triangular system are $3,-6,16$ .

Explanation of Solution

Given:

Equation’s given are,

  $x+2y+z=$ $7$ .... $\left(1\right)$

  $-y+3z=$ $9$ .... $\left(2\right)$

  $2z=$ $6$ .... $\left(3\right)$

Concept Used:

Back substitution is used to find the value of $x,y,z$.

Calculation:

Consider the given equations

  $x+2y+z=$ $7$ .... $\left(1\right)$

  $-y+3z=$ $9$ .... $\left(2\right)$

  $2z=$ $6$ .... $\left(3\right)$

The main purpose is to solve the equations using back substitution, From equation $\left(3\right)$ the value of $z$ can be obtained.

  $2z=$ $6$ .... $\left(3\right)$

  $\begin{array}{l}z=\frac{6}{2}=3\\ z=\boxed{3}\end{array}$

Now from equation $\left(2\right)$ to find the value of $y$

  $-y+3z=$ $9$ .... $\left(2\right)$

Substituting value of $z$ in equation $\left(2\right)$ and solve,

  $\begin{array}{l}-y+3*2=9\\ -y+6=9\\ -y=9-6\\ -y=6\end{array}$

  $\boxed{y=-6}$

Now, to find the value of $x$ , From equation $\left(1\right)$

  $x+2y+z=$ $7$ .... $\left(1\right)$

Substitute the values of $y$ and $z$ in equation $\left(1\right)$ and solve,

  $\begin{array}{l}x+2*\left(-6\right)+3=7\\ x-12+3=7\\ x-9=7\\ x=7+9\end{array}$

  $\boxed{x=16}$

Conclusion:

Hence, the values $x,y,z$using back substitution of triangular system are $3,-6,16$ .