Problem 68 - Elementary Algebra

Question

For the following problems, find the slope of the line through the pairs of points. $$ (4,2),(6,2) $$

Step-by-Step Solution

Answer

Answer: The slope of the line is 0.

Step 1: Identify the coordinates of the points

We have been given two points, (4, 2) and (6, 2). The coordinates for the points are: $$ x_{1} = 4, \ y_{1} = 2 \\ x_{2} = 6, \ y_{2} = 2 $$

Step 2: Plug the coordinates into the slope formula

Now, we will use the coordinates of the two points in the slope formula: $$ m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \\ m = \frac{2 - 2}{6 - 4} $$

Step 3: Calculate the slope

Solve for m: $$ m = \frac{2 - 2}{6 - 4} \\ m = \frac{0}{2} \\ m = 0 $$ The slope of the line passing through the points (4, 2) and (6, 2) is 0.

Key Concepts

Understanding Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a mathematical concept that employs algebraic equations to represent geometric shapes and figures on a coordinate plane. This form of geometry allows us to explore geometric properties and relationships using a pair of coordinates, usually referred to as

• \((x, y)\) in two-dimensional space.

These coordinates are critical for finding key information about lines, such as their slope, length, and intersection points.
By understanding coordinate geometry, we can easily map out points in space and determine relationships and features of the lines that connect them. It offers a bridge between abstract spaces and tangible solutions to problems involving spatial figures.

Explaining the Slope Formula

The slope of a line is a measure of how steep the line is. To calculate the slope when given two points, we use the slope formula. You'll often see it expressed as:

• \(m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\)

Here, \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) are the coordinates of two distinct points on the line.
The slope \(m\) represents the "rise over run," which is the ratio of the change in the y-coordinates (vertical change) to the change in the x-coordinates (horizontal change). This formula is incredibly useful because it gives a clear and concise way to determine the direction and steepness of a line connecting any two points on a coordinate plane.
Utilizing this precise formula makes it easier to analyze and graph linear equations and functions.

Recognizing a Horizontal Line

A horizontal line is a straight line that runs from left to right or right to left on a coordinate plane, remaining at a constant elevation.
One key characteristic of horizontal lines is that all points on the line have the same y-coordinate. For the given exercise, the points are \((4, 2)\) and \((6, 2)\), showcasing that the y-values do not change.

• In a slope formula, a horizontal line will have \(y_{2} - y_{1} = 0\).

This calculation explains why the slope of a horizontal line is always 0.Understanding this distinctive nature of horizontal lines helps to visualize and differentiate them easily from other types of lines, such as vertical lines (which have undefined slopes) and diagonal lines (which have defined non-zero slopes). Recognizing this, you can anticipate their behavior in various mathematical problems and real-world scenarios involving analytics and measurements.