Problem 7 - 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. $$(-2,4) \text { and }(-1,-1)$$

Step-by-Step Solution

Answer

The slope of the line is -5. Therefore, the line is falling.

Step 1: Identify the coordinates

The given points are (-2,4) and (-1,-1). Here, the first point is \( (x_1, y_1) = (-2,4) \) and the second point is \( (x_2, y_2) = (-1,-1) \).

Step 2: Plug the values into the slope formula

Now, use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \) to find the slope. Plug the coordinates of the points into the formula to get \[ m = \frac{-1 - 4}{-1 - (-2)} = -5 \]. The slope is -5.

Step 3: Determine the behavior of the line

The slope is negative, which means the line is falling. A negative slope means we go down from left to right.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that utilizes algebraic equations to represent geometric principles and problems. It especially comes in handy when examining points, lines, and shapes within the Cartesian coordinate system.

In this system, we work with a two-dimensional grid defined by an x-axis (horizontal) and a y-axis (vertical). Each point on the grid is identified by an ordered pair of numbers known as coordinates, which indicate its position relative to the origin (0,0). The first number, or x-coordinate, determines how far to move horizontally from the origin, while the second, the y-coordinate, tells us the vertical displacement.

Understanding coordinate geometry is essential for solving problems that involve graphing lines and figuring out the intricate relationships between geometric figures and algebraic expressions.

Slope Formula

The slope of a line in coordinate geometry is a measure of its steepness and direction, and it's commonly denoted by the letter 'm'. To calculate the slope between two points, \( (x_1, y_1) \) and \( (x_2, y_2) \), we use the slope formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

This formula essentially calculates the change in the y-values (vertical change) divided by the change in the x-values (horizontal change). When looking at individual points, for instance, \( (-2,4) \) and \( (-1,-1) \), plugging in these coordinates will help determine the line's slope.

It is critical to follow the order of the coordinates as switching them may result in a different sign for the slope. A positive slope indicates that the line rises from left to right, whereas a negative slope means the line falls in that direction.

Line Behavior Analysis

Analyzing the behavior of a line involves determining whether it has a positive slope, a negative slope, or no slope at all - which would indicate a horizontal or vertical line.

A line with a positive slope rises from left to right. This upward tilt correlates with a positive relationship between the x and y values — as x increases, so does y. Conversely, a line with a negative slope, such as the one with a slope of -5 in our exercise, falls as it moves from left to right. This indicates a negative relationship, meaning as x values increase, y values decrease.

There are also special cases. A slope of zero signifies a horizontal line — it does not rise or fall regardless of the x-value changes. If the slope is undefined, which occurs when \( x_2 = x_1 \), the line is vertical, as it does not run across the x-axis at all.

Through this line behavior analysis, we can better understand the graphical representation of linear equations and how they reflect various types of quantitative relationships.