Problem 49 - Elementary Algebra Within Reach

Question

In Exercises 49-54, plot the points and find the slope (if possible) of the line that passes through the points. If not possible, state why. $$ (-6,4),(-3,-4) $$

Step-by-Step Solution

Answer

The slope of the line that passes through the points (-6,4) and (-3,-4) is \( \frac{-8}{3} \).

Step 1: Plot the Points

Plot the points (-6,4) and (-3,-4) onto a graph. It's good to put these points on the graph to visualize better.

Step 2: Construct a Line

Draw a line across the two points. This line represents the slope we need to find. If the line is perfectly vertical or horizontal, it indicates an undefined or zero slope.

Step 3: Find the Slope

Use the formula for finding the slope of a line which is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting the coordinates of the given points, we get \( m = \frac{-4 - 4}{-3 - (-6)} = \frac{-8}{3} \).

Key Concepts

Slope of a Line

The slope of a line is a measure of its steepness or inclination. In algebra, understanding the slope is crucial as it tells us how one variable changes in relation to another. To determine the slope between two points, we use the slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two points on the line.
This formula calculates the vertical change ("rise") over the horizontal change ("run") between the points. The slope can be positive, negative, zero, or undefined.

• A positive slope indicates that the line rises as we move from left to right.
• A negative slope indicates that the line falls as we move from left to right.
• A zero slope signifies a horizontal line.
• An undefined slope indicates a vertical line where the change in x is zero.

Understanding the slope will help in interpreting graphs and analyzing linear relationships.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves studying geometric figures through a coordinate system. In this system, points are defined by their positions on the x (horizontal) and y (vertical) axes. Each point is represented as an ordered pair \((x, y)\), allowing us to perform algebraic operations to solve geometric problems.
This branch of geometry makes it easier to visualize and solve geometry problems by using the coordinates instead of just abstract figures.
Some fundamental uses include:

• Calculating the distance between two points.
• Finding the midpoint of a line segment.
• Determining slopes of lines, which in turn helps to find equations of the lines.

Coordinate geometry is essential in connecting algebra with geometry, providing a practical system to describe geometric shapes and their properties.

Plotting Points

Plotting points is a simple yet vital skill in coordinate geometry. It involves placing points on a graph based on their coordinates \((x, y)\). The x-coordinate tells how far to move horizontally from the origin, and the y-coordinate tells how far to move vertically.
This forms the basis of graphing linear equations and analyzing graphs.

• Start with the x-coordinate: Move left for negative values, right for positive.
• Next, the y-coordinate: Move up for positive values, down for negative.
• Mark the point where these movements intersect.

Practicing plotting points helps improve spatial awareness and understanding of graphs. This process is critical for accurately representing and analyzing data, functions, and various algebraic equations.