Chapter 9, Problem 1R

Draw the graphs of each pair of linear equations on the same set of coordinate axes. Find the point of intersection. If the lines do not intersect, tell whether the lines are parallel or coincide:

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

To determine

To graph: The pair of linear equations:

$\begin{array}{c}x+y=6\\ 2x-y=3\end{array}$

And find the point of intersection. If line do not intersect, then tell whether the lines are parallel or coincide.

Answer to Problem 1R

Solution:

The point of intersection is $\underset{\_}{\left(3,3\right)}$.

Explanation of Solution

Given Information:

The pair of linear equations:

$\begin{array}{c}x+y=6\\ 2x-y=3\end{array}$

Graph:

Consider the pair of linear equations:

$\begin{array}{c}x+y=6\\ 2x-y=3\end{array}$

To graph it first consider the equation $x+y=6$,

That is $y=6-x$.

Now plot the different ordered pairs for this equation on the graph.

For $x=0$ the value of y is:

$\begin{array}{c}y=6-0\\ =6\end{array}$

For $x=3$ the value of y is:

$\begin{array}{c}y=6-3\\ =3\end{array}$

For $x=-3$ the value of y is:

$\begin{array}{c}y=6-\left(-3\right)\\ =6+3\\ =9\end{array}$

The table for the given equation is:

$\begin{array}{cccc}x& 0& 3& -3\\ y& 6& 3& 9\end{array}$

Now for the equation $2x-y=3$,

The value of y can be represented as $y=2x-3$.

Now plot the different ordered pairs for this equation on the graph.

For $x=0$ the value of y is:

$\begin{array}{c}y=2\left(0\right)-3\\ =-3\end{array}$

For $x=3$ the value of y is:

$\begin{array}{c}y=2\left(3\right)-3\\ =6-3\\ =3\end{array}$

For $x=-3$ the value of y is:

$\begin{array}{c}y=2\left(-3\right)-3\\ =-6-3\\ =-9\end{array}$

The table for the given equation is:

$\begin{array}{cccc}x& 0& 3& -3\\ y& -3& 3& -9\end{array}$

The graph of both the equations is:

From the graph it can be seen that both the lines will intersect at $\left(3,3\right)$.

Interpretation:

The graph shows that the point of intersection of both the equations is $\left(3,3\right)$. This means at this point the value of both the equations will be the same.