Chapter 10.2, Problem 7E

To determine

Using back substitution solve the triangular system.

Answer to Problem 7E

The values $x,y,z$using back substitution of triangular system are $2,3,11$ .

Explanation of Solution

Given:

Equation’s given are,

  $x-2y+4z=3$

  $y+2z=7$

  $z=2$

Concept Used:

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

Calculation:

Consider the given equations

  $x-2y+4z=3$

  $y+2z=7$

  $z=2$

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

  $\boxed{z=2}$

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

  $y+2z=7$

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

  $\begin{array}{l}y+2z=7\\ y+2\times 2=7\\ y+4=7\end{array}$

Subtracting $4$ from $7$ to get $y$

  $\begin{array}{l}y=7-4\\ \boxed{y=3}\end{array}$

Now, to find the value of $x$ ,

From equation $\left(1\right)$

  $x-2y+4z=3$

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

  $\begin{array}{l}x+2*3+4*2=3\\ x+6+8=3\\ x+14=3\end{array}$

Subtracting $3$ from $14$ to get $x$

  $x=14-3$

  $\boxed{x=11}$

Conclusion:

Hence, the values $x,y,z$using back substitution of triangular system are $2,3,11$ .