Chapter 7.2, Problem 49E

To determine

To graph : the line in the system and determine whether the system is consistent or not.

Explanation of Solution

Given information :

  $\{\begin{array}{l}6x+3y=-8\\ -x-\frac{1}{2}y=\frac{4}{3}\end{array}$

Graph:Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the expression ${\text{Y}}_1\text{=6}x+3y=-8$ and ${\text{Y}}_2\text{=}-x-\frac{1}{2}y=\frac{4}{3}$ which is required to graph.

Step 4: Press GRAPH button to graph the function.

The graph is obtained as:

  

The system is consistent and dependent because the line of given equationdidn’t coincideeach other and it has infinite number of point in common

To check the solution solved both the equation algebraically is obtained as:

Let, $\text{6}x+3y=-8$ is the equation (i) and $-x-\frac{1}{2}y=\frac{4}{3}$ is the equation (ii).

  $\begin{array}{l}\underset{\_}{\begin{array}{l}\text{6}x+3y=-8\\ -6x-6y=8\end{array}}\left[\text{multiply 6 in eqaution }\left(\text{ii}\right)\right]\\ \text{ 0}=0\end{array}$

Because $\text{ 0}=0$ for all the values of $x$ and $y$ ,the two equation turn to be equivalent .

Hence, the system has infinitely many solutions.

Interpretation : From the above graph, it can be observed thatthe system is consistent and dependent because the line of given equation didn’t coincide each other and it has infinite number of point in common