Problem 6 - Introductory Algebra for College Students

Question

In Exercises \(1-10,\) 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. $$(4,-1) \text { and }(3,-1)$$

Step-by-Step Solution

Answer

The slope of the line passing through the points (4, -1) and (3, -1) is 0. The line is horizontal.

Step 1: Identify the coordinates of the points

Identify the coordinates of the given points. The first point is (4, -1), so \(x_1 = 4\), \(y_1 = -1\). The second point is (3, -1), so \(x_2 = 3\), \(y_2 = -1\).

Step 2: Calculate the slope

Use the slope formula \( m= \frac{{y_2-y_1}}{{x_2-x_1}}\) to find the slope. Substituting the given values in the formula: \( m = \frac{{-1 - -1}}{{3-4}} \). Simplifying this we get \( m = 0 \).

Step 3: Determine the condition of the line

Since the calculated slope is 0, the line is horizontal.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to describe and investigate geometric shapes and figures. In essence, it provides a connection between geometry and algebra through graphs involving coordinates.

Let's start with the basics of a coordinate system. A two-dimensional Cartesian coordinate system consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). The point where these axes intersect is known as the origin, designated as (0,0). Every point in the plane can be specified by an ordered pair of numbers, (x, y), known as coordinates. The first number, 'x', represents the horizontal position relative to the origin, while the second number, 'y', represents the vertical position.

In coordinate geometry, the position and shape of a geometric figure, like a line, can be described by equations or plotted using these pairs of numbers. For the case of lines, these equations often take the form y = mx + b, where 'm' represents the slope of the line and 'b' represents the y-intercept, the point at which the line crosses the y-axis.

Slope Formula

Understanding the slope of a line is a fundamental concept in coordinate geometry. The slope indicates the steepness and the direction of a line. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

The typical formula to calculate the slope, denoted as 'm', is

Slope Formula:

 But this mathematical procedure is more than just a calculation; it’s a powerful tool for analyzing the behavior of lines. By using the slope formula, we can predict how a line will look on a graph and understand the relationship it conveys between x and y values.

For example, in the problem presented, calculating the slope of a line passing through the points (4, -1) and (3, -1) involves finding the difference in the y-coordinates and dividing by the difference in the x-coordinates. By applying the slope formula effectively, we determine whether the line rises or falls, or if it is a special case, such as horizontal or vertical.

Horizontal Line

Horizontal lines are a special type of line in coordinate geometry. They have unique properties that set them apart from other lines. A horizontal line has a constant y-coordinate, meaning it doesn't rise or fall as it moves left to right or vice versa. As a result, its slope is zero.

In the given exercise, after applying the slope formula, we concluded that the slope (m) of the line through the points (4, -1) and (3, -1) is 0. This indicates that there is no vertical change between the two points — they have the same y-coordinate. Therefore, the line is perfectly horizontal.

A horizontal line is described by an equation of the type y = c, where 'c' is the constant y-coordinate of the line. This simplicity makes horizontal lines easy to identify on a graph. They are useful references in various applications, from architecture to graph interpretation in statistics. Knowing a line is horizontal helps to predict and understand its relationship within a coordinate plane, particularly that it will not intersect with other horizontal lines unless they are the same line.