Problem 9 - College Algebra Essentials

Question

Find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises falls, is horizontal, or is vertical. $$(5,3) \text { and }(5,-2)$$

Step-by-Step Solution

Answer

The slope of the line passing through (5,3) and (5,-2) is undefined, and the line is vertical.

Step 1: Calculating the slope

Substitute the given points into the slope formula \( m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \). In this case, \( (x_{1}, y_{1}) = (5,3) \) and \( (x_{2}, y_{2}) = (5,-2) \), so the initial equation is \( m = \frac{-2 - 3}{5 - 5} \). The denominator of this formula is zero, which makes m undefined. The slope is thus not calculated as it would require division by zero.

Step 2: Determine the direction of the line

Since the denominator of the slope equation, which represents the difference in x coordinates, is zero, this indicates that both points lie on a vertical line. Therefore, the line is vertical.

Key Concepts

Vertical Line

A vertical line is a significant concept in geometry. It is a straight line that goes up and down instead of left to right. One of the main characteristics of a vertical line is that all the points on it have the same x-coordinate. In our previous exercise, the points (5,3) and (5,-2) both have x-values of 5. This confirms that they align perfectly on a vertical line.
Vertical lines have unique properties, such as:

• They can run forever in both directions without bending or deviating.
• They are perpendicular to horizontal lines.
• They visually depict the same x-value for different y-values.

Understanding the distinctiveness of vertical lines helps in recognizing line orientations in geometry.

Undefined Slope

The term "undefined slope" arises when attempting to calculate the slope of a vertical line. The slope of a line is the measure of its steepness, often calculated using the slope formula. For our vertical line, when we plug the coordinates (5,3) and (5,-2) into the formula, the denominator becomes zero:
\[ m = \frac{-2 - 3}{5 - 5} = \frac{-5}{0} \]
Division by zero is mathematically undefined, which means the slope cannot be calculated for vertical lines. Hence, we say the slope is "undefined."
Vertical lines defy the typical slope calculability, distinguishing them from non-vertical lines.

Coordinates

Coordinates are essential in determining the position of points in a plane. They are usually written as pairs of numbers, such as (x, y). The x-coordinate tells us how far, and in which direction, the point is from the vertical y-axis. Meanwhile, the y-coordinate gives us the distance and direction from the horizontal x-axis.
In your exercise, the coordinates (5,3) and (5,-2) represent specific locations in a plane.

• Both have the same x-coordinate, 5, placing them vertically aligned.
• The y-coordinates differ, indicating their separation distance along the vertical axis.

Grasping coordinate principles is crucial in plotting points and understanding their geometric relationship.

Slope Formula

The slope formula is a fundamental tool for calculating how slanted a line is. It is expressed as:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
This formula computes the "rise over run," where "rise" is the change in y-values, and "run" is the change in x-values. The concept is simple: it tells us how much the line goes up or down for a given horizontal movement.
In the case of vertical lines, like with points (5,3) and (5,-2), the difference in x-values will equal zero, because they share the same x-coordinate. Thus, the slope becomes undefined, depicting that the line does not "run" horizontally at all.
Through this understanding, the slope formula aids in identifying not only the steepness of lines but also their type, including horizontal or vertical orientations.