Chapter 7.2, Problem 50E

To determine

To graph : the line in the system and check whether the system is consistent or inconsistent.

Explanation of Solution

Given information :

  $\{\begin{array}{l}-\frac{1}{4}x-\frac{1}{2}y=1\\ 5x+y=1\end{array}$

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 ${\text{Y}}_1\text{=}-\frac{1}{4}x-\frac{1}{2}y=1$ and ${\text{Y}}_2\text{=}5x+y=1$ which is required to graph.

Step 4: Press GRAPH button to graph the function.

The graph is obtained as:

  

The system is consistent because the line of the given equation intersect each other and the point of intersection of both line is $x=\frac{2}{3}$ and $y=-\frac{7}{3}$ .

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

Let, $-\frac{1}{4}x-\frac{1}{2}y=1$ is the equation (i) and $5x+y=1$ is the equation (ii).

  $\begin{array}{l}\underset{\_}{\begin{array}{l}-5x-10y=20\\ 5x+y=1\end{array}}\left[\text{multiply 20 in equation (i) }\right]\\ -9y=21\left[\text{add equation}\right]\\ y=\frac{21}{-9}\\ y=-\frac{7}{3}\end{array}$

From this equation, substitute the value of $y=-\frac{7}{3}$ into equation (ii).

  $\begin{array}{l}5x+y=1\\ 5x-\frac{7}{3}=1\left[\text{substitute }\right]\\ 5x=1+\frac{7}{3}\\ 5x=\frac{10}{3}\\ x=\frac{10}{3\cdot 5}\\ x=\frac{2}{3}\end{array}$

Hence, the solution of linear equation is $x=\frac{2}{3}$ and $y=-\frac{7}{3}$ .

Interpretation : From the above graph, it can be observed thatthe system is consistent because the line of given equation intersect each other and the solution of the system is $x=\frac{2}{3}$ and $y=-\frac{7}{3}$ .