Chapter 4, Problem 1RP

Solve the following systems by graphing. $\begin{array}{l}x+2y=8\\ x-y=2\end{array}$

To determine

To calculate: The solution of the following equations by graphing method.

$\begin{array}{ccc}\hfill x+2y& =& 8\hfill \\ \hfill x-y& =& 2\hfill \end{array}$

Answer to Problem 1RP

Solution:

The graphs of both the equations intersect at the point $\underset{\_}{\left(4,2\right)}$.

Explanation of Solution

Given:

The set of equations is:

$\begin{array}{ccc}\hfill x+2y& =& 8\hfill \\ \hfill x-y& =& 2\hfill \end{array}$

Calculation:

The graphs for the equations are made by constructing tables for the equations with values of x and y.

Consider the following equation:

$x+2y=8$

When $x=0$,

Substitute 0 for x.

$\begin{array}{ccc}\hfill \left(0\right)+2y& =& 8\hfill \\ \hfill 2y& =& 8\hfill \\ \hfill y& =& \frac{8}{2}\hfill \\ \hfill y& =& 4\hfill \end{array}$

When $x=2$,

Substitute 2 for x.

$\begin{array}{ccc}\hfill \left(2\right)+2y& =& 8\hfill \\ \hfill 2y& =& 8-2\hfill \\ \hfill y& =& \frac{6}{2}\hfill \\ \hfill y& =& 3\hfill \end{array}$

When $x=-2$,

Substitute $-2$ for x.

$\begin{array}{ccc}\hfill \left(-2\right)+2y& =& 8\hfill \\ \hfill 2y& =& 8+2\hfill \\ \hfill y& =& \frac{10}{2}\hfill \\ \hfill y& =& 5\hfill \end{array}$

Construct a table for the various values of x and y for the equation, $x+2y=8$.

x	0	2	$-2$
y	4	3	5

Consider the another equation.

$x-y=2$

When $x=0$,

Substitute 0 for x.

$\begin{array}{ccc}\hfill \left(0\right)-y& =& 2\hfill \\ \hfill -y& =& 2\hfill \\ \hfill y& =& -2\hfill \end{array}$

When $x=1$,

Substitute 1 for x.

$\begin{array}{ccc}\hfill \left(1\right)-y& =& 2\hfill \\ \hfill -y& =& 2-1\hfill \\ \hfill y& =& -1\hfill \end{array}$

When $x=4$,

Substitute 4 for x.

$\begin{array}{ccc}\hfill \left(4\right)-y& =& 2\hfill \\ \hfill -y& =& 2-4\hfill \\ \hfill y& =& 2\hfill \end{array}$

Construct a table for the various values of x and y for the equation, $x-y=2$.

x	0	1	4
y	$-2$	$-1$	2

Graph:

Recall the table obtained for the equations:

$x+2y=8$

x	0	2	$-2$
y	4	3	5

And,

$x-y=2$

x	0	1	4
y	$-2$	$-1$	2

Plot the graph using the values of both the tables is:

Interpretation:

The graphs of both the equations intersect at the point $\left(4,2\right)$. Thus, the system of equation has unique solution.