Chapter 8.5, Problem 33E

a.

To determine

To show:

Two points on the graph of $y=mx+b$ are $(0,b)\text{and }(1,m+b)$ . Draw the line that appears to best fit the data points.

Explanation of Solution

Given:

The slope of the line $y=mx+b$ is m .

Calculation:

To solve our given problem, we will find the values of y , when $x=0\text{and when }x=1$ as shown below:

  $\begin{array}{l}y=mx+b\\ y=m(0)+b\\ y=0+b\\ y=b\end{array}$

  $\begin{array}{l}y=mx+b\\ y=m(1)+b\\ y=m+b\end{array}$

We can see that the value of y is b, when $x=0$ and value of y is $m+b$ at $x=1$ . So points $(0,b)\text{and }(1,m+b)$ are on the graph of $y=mx+b$ .

Hence proved.

b.

To determine

For points $(0,b)\text{and }(1,m+b)$ , what is the difference of the second y -coordinate and the first y -coordinate? What is the difference of the second x -coordinate and the first x -coordinate?

Answer to Problem 33E

The difference of the second y -coordinate and the first y -coordinate is m .

The difference of the second x -coordinate and the first x -coordinate is 1.

Explanation of Solution

Given:

The slope of the line $y=mx+b$ is m .

Calculation:

Let us find the difference of the second y -coordinate and the first y -coordinate by subtracting 1^st coordinate from 2^nd coordinate as:

  $\begin{array}{l}\text{Difference between }y\text{-coordinates}=m+b-b\\ \text{Difference between }y\text{-coordinates}=m\end{array}$

Now we will find the difference of the second x- coordinate and the first x -coordinate by subtracting 1^st coordinate from 2^nd coordinate as:

  $\begin{array}{l}\text{Difference between }x\text{-coordinates}=1-0\\ \text{Difference between }x\text{-coordinates}=1\end{array}$

Therefore, the difference of the second y -coordinate and the first y -coordinate is m and the difference of the second x -coordinate and the first x -coordinate is 1.

c.

To determine

To write:

An expression for the slope of the line $y=mx+b$ using your results from part (b). Show that the slope is equal to m .

Explanation of Solution

Given:

The slope of the line $y=mx+b$ is m .

Calculation:

We know that the slope of a line is rise over run. The rise is difference between the y -coordinates of two points on a line, while the run is difference between the x -coordinates of two points on a line.

  $\begin{array}{l}\text{Slope}=\frac{\text{Rise}}{\text{Run}}\\ \text{Slope}=\frac{m}{1}\\ \text{Slope}=m\end{array}$

Hence proved.