Chapter 10.2, Problem 10E

To determine

Using back substitution solve the triangular system.

Answer to Problem 10E

The values $x,y,z$using back substitution of triangular system are $4,3,16$ .

Explanation of Solution

Given:

Equation’s given are,

  $x-2y+3z=$ $10$ …. $\left(1\right)$

  $2y-z=$ $2$ …. $\left(2\right)$

  $3z=$ $12$ …. $\left(3\right)$

Concept Used:

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

Calculation:

Consider the given equations

  $x-2y+3z=$ $10$ …. $\left(1\right)$

  $2y-z=$ $2$ …. $\left(2\right)$

  $3z=$ $12$ …. $\left(3\right)$

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

  $3z=$ $12$ …. $\left(3\right)$

  $\begin{array}{l}z=\frac{12}{3}=4\\ z=\boxed{4}\end{array}$

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

  $2y-z=$ $2$ …. $\left(2\right)$

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

  $2y-z=$ $2$

  $\begin{array}{l}2y=2+4\\ 2y=6\\ y=\frac{6}{2}\end{array}$

  $y=\boxed{3}$

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

  $x-2y+3z=$ $10$ …. $\left(1\right)$

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

  $x-2y+3z=$ $10$

  $\begin{array}{l}x-2*3+3*4=10\\ x-6+12=10\\ x-6=10\\ x=10+6\\ x=\boxed{16}\end{array}$

Conclusion:

Hence, the values $x,y,z$using back substitution of triangular system are $4,3,16$ .