Chapter D, Problem 1E

To determine

(a)

To find:

The point of intersection of the equations $y=3x+2$ and $y=5x+5$

Answer to Problem 1E

Solution:

The point of intersection of the equations $y=3x+2$ and $y=5x+5$ is $\left(-2.5,-2.5\right)$.

Explanation of Solution

Calculation:

The steps to find the point of intersection using a graphing calculator are follows,

• Set the calculator up for instructions on what to graph by pressing $\boxed{Y=}$. This produces the screen as follows.

Figure $\left(1\right)$

• Type “$3x+2$” where the screen reads $\boxed{Y_1=}$ and type “$5x+5$” where the screen reads $\boxed{Y_2=}$. To type the symbol $x$ press $\boxed{X.T.\theta .n}$. After typing “$3x+2$” and “$5x+5$” press the $\boxed{\text{ENTER}}$ button. This produces the following screen.

Figure $\left(2\right)$

• Press the $\boxed{\text{WINDOW}}$ button to graph the equation by entering the values as shown below. “Xmin” and “Xmax” refers to the left and right boundaries of the graph, respectively. “Ymin” and “Ymax” refers to the lower an upper boundary of the graph, respectively. “Xscl” and “Yscl” refer to the scales on the x and y axes.

Figure $\left(3\right)$

• Press $\boxed{\text{2nd}}$$\boxed{\text{CALC}}$ button and select option 5. This produces the following screen.

Figure $\left(4\right)$

• When the calculator responds with first curve and a mark on the first equations graph, press $\boxed{\text{ENTER}}$.

Figure $\left(5\right)$

• When the calculator respond with and a mark on the first equations graph, press $\boxed{\text{ENTER}}$.

Figure $\left(6\right)$

• When the calculator responds with guess then use left and right arrows to place the mark near the points of intersection and press $\boxed{\text{ENTER}}$. The calculator will display the location of intersection.

Figure $\left(7\right)$

The point of intersection of the equations $y=3x+2$ and $y=5x+5$ is shown in the figure above.

Therefore, the point of intersection of the equations $y=3x+2$ and $y=5x+5$ is $\left(-2.5,-2.5\right)$

Conclusion:

Hence, the point of intersection of the equations $y=3x+2$ and $y=5x+5$ is $\left(-2.5,-2.5\right)$

To determine

(b)

To check:

The solution by substituting the ordered pair into each of the two equations.

Answer to Problem 1E

Solution:

The point of intersection calculated by the graphing calculator is correct.

Explanation of Solution

Given:

The equation of lines is,

$y=3x+2$

$y=5x+5$

Approach:

Substitute $-2.5$ for $x$ in both the equation and check the value of $y$.

Calculation:

Consider the first equation.

$y=3x+2$

Substitute $-2.5$ for $x$ in the above equation.

$\begin{array}{rll}y& =& 3\left(-2.5\right)+2\\ & =& -4.5+2\\ & =& -2.5\end{array}$

Consider the second equation.

$y=5x+5$

Substitute $-2.5$ for $x$ in the above equation.

$\begin{array}{rll}y& =& 5\left(-2.5\right)+5\\ & =& -7.5+5\\ & =& -2.5\end{array}$

Since the value of $x$ and $y$ is $\left(-2.5,-2.5\right)$ the point of intersection calculated by the graphing calculator is correct.

Conclusion:

Hence, the point of intersection calculated by the graphing calculator is correct.