Chapter 3.3, Problem 51HP

To determine

To explain how the rate of change and slope are related and how to find the slope of a line.

Explanation of Solution

Concept Used:

Equation of the line passing through the points $\left(x_1,y_1\right)\text{ and }\left(x_2,y_2\right)$ is given by $y-y_1=m\left(x-x_1\right)$ , where $\text{slope, }m=\frac{y_2-y_1}{x_2-x_1}$ .

Calculation:

The slope of any line is given by rise over run. That is, the ratio of the number of units it rise or fall over the number of units it move forward or backward from a fixed point.

The rate of change is a ratio to describe how much one quantity is changing with respect to a change in another quantity.

The slope of the line is the instantaneous rate of change at any point.

The slope of a line can be found using rise over run method. In this method, start from any point on the line and then find the ratio of the count the units moved up/down to the count of units moved forward/backward. If the units measured for going upward or forward then take the units with positive sign. And if the units measured are in downward or backward direction, then take the number of units with negative sign.

For example, the slope of the line passing through the points $\left(a,b\right)\text{ and }\left(c,d\right)$ is given by

  $\begin{array}{c}\text{slope, }m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{d-b}{c-a}\end{array}$