Question

Triangular Systems

Use back-substitution to solve the triangular system

$\left\{\begin{array}{l}x+2y+z=7\\ -y+3z=9\\ 2z=6\end{array}\right.$

Short answer

Hence we get the solution of the triangular system as

$\begin{array}{l}x=4\\ y=0\\ z=3\end{array}$

Step-by-step solution

Step 1. Given information

We are given the set of linear equations that is-

$\left\{\begin{array}{l}x+2y+z=7\\ -y+3z=9\\ 2z=6\end{array}\right.$

Step 2. Concept used

The process of solving a linear system of equations that has been transformed into the row-echelon form or reduced row-echelon form is known as the back-substituting method.

We have to solve the given triangular system by using the back substitution method.

Step 3. Calculation

We have the triangular systems:

$\begin{array}{l}x+2y+z=7\\ -y+3z=9\\ 2z=6\end{array}$

Since we have

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

Now substituting the value of $z=3$ in the equation $-y+3z=9$, we get

$\begin{array}{l}-y+3\left(3\right)=9\\ y=9-9\\ y=0\end{array}$

Now using the values of $y,z$, we will solve the equation $x+2y+z=7$

We get

$\begin{array}{l}x+2\left(0\right)+3=7\\ x=7-3\\ x=4\end{array}$

So, we solve the triangular system.