Problem 23 - Technical Mathematics with Calculus

Question

Slope Find the slope of each straight line. Rise \(=-9 ;\) run \(=-3\)

Step-by-Step Solution

Answer

The slope of the line is 3.

Step 1: Understanding slope

Slope is defined as the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. It is often represented as 'm'.

Step 2: Calculating the slope

To find the slope with given rise and run, use the formula slope (m) = rise/run. Substitute -9 for rise and -3 for run into the formula.

Step 3: Simplifying the fraction

Divide -9 by -3 to simplify the fraction, which will give you the slope of the line.

Key Concepts

Calculating Slope

Understanding how to calculate slope is essential when studying linear equations and graphing lines. The slope of a line is a numerical representation of the line's steepness and direction. It quantifies how much a line rises or falls as it moves from left to right on the Cartesian plane.

Starting with simple cases, when we are provided with the amount a line rises vertically and the amount it spans horizontally between any two points, we can easily calculate the slope. This straightforward approach is particularly useful when we work with right triangles formed between the points on a graph and can identify the lengths of their sides.

Ratio of Rise Over Run

The concept 'rise over run' is pivotal to understanding slope. Imagine you're climbing or descending stairs; the 'rise' refers to the vertical change you experience, while the 'run' refers to your horizontal movement. In mathematical terms, this analogy translates to the change in the y-coordinates (rise) and the change in the x-coordinates (run) of two points on the line.

In our everyday life, this ratio represents how inclined or declined a ramp, a hill, or any path is. In mathematics, this ratio is what we call the slope of a line in a coordinate plane. It can have positive, negative, zero, or undefined values, each representing different characteristics of the line. A positive slope ascends from left to right, a negative slope descends, a zero slope is perfectly horizontal, and an undefined slope is vertical.

Slope Formula

The slope formula, stated as \( m = \frac{rise}{run} \), is a valuable tool for finding the steepness and direction of a line. To employ this formula, you first identify two points on the line and calculate the differences in their y-coordinates and x-coordinates, which represent the 'rise' and 'run', respectively.

Using the exercise provided as an example:

Rise = -9

Run = -3

When these values are substituted into the slope formula, we get \( m = \frac{-9}{-3} \). Simplifying the fraction gives us the slope of the line, which in this case would be positive 3. This positive slope indicates that the line rises as it moves from left to right. It's crucial to carefully deal with the signs of the rise and run since they determine the line's direction.