Problem 37 - Precalculus with Limits

Question

In Exercises 29-40, plot the points and find the slope of the line passing through the pair of points. \( (\frac{11}{2}, -\frac{4}{3}) \), \( (-\frac{3}{2}, -\frac{1}{3}) \)

Step-by-Step Solution

Answer

The slope of the line passing through the points \( (\frac{11}{2}, -\frac{4}{3}) \) and \( (-\frac{3}{2}, -\frac{1}{3}) \) is -\frac{1}{7}.

Step 1: Identify coordinates

The given points are \( (\frac{11}{2}, -\frac{4}{3}) \) and \( (-\frac{3}{2}, -\frac{1}{3}) \). So, \( x_1 = \frac{11}{2}, y_1 = -\frac{4}{3}, x_2 = -\frac{3}{2}, y_2 = -\frac{1}{3} \).

Step 2: Calculate the slope

Substitute these coordinates into the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). So, the slope \( m = \frac{-\frac{1}{3} - (-\frac{4}{3})}{-\frac{3}{2} - \frac{11}{2}} = \frac{1}{-7} = -\frac{1}{7} \).

Step 3: Plot the points

Plot the coordinates \( (\frac{11}{2}, -\frac{4}{3}) \) and \( (-\frac{3}{2}, -\frac{1}{3}) \) on a graph where x-axis represents x-coordinates and y-axis represents y-coordinates.

Key Concepts

Plotting Points

Plotting points on a graph is a fundamental skill in mathematics. To get started, you need a coordinate plane, which consists of a horizontal line called the x-axis and a vertical line called the y-axis. These axes intersect at the origin point, which is (0,0). Points are plotted by identifying their coordinates, which are typically written in the form of (x, y). The x-coordinate tells us how far to move horizontally from the origin, while the y-coordinate tells us how far to move vertically.

Here’s a step-by-step guide to plotting the points in the given exercise:

• Identify the first point: \( (\frac{11}{2}, -\frac{4}{3}) \).
• The x-coordinate \( \frac{11}{2} \) means you move 5.5 units to the right of the origin.
• The y-coordinate \( -\frac{4}{3} \) means you move approximately 1.33 units down from there.
• Plot the point on the graph.
• Identify the second point: \( (-\frac{3}{2}, -\frac{1}{3}) \).
• The x-coordinate \( -\frac{3}{2} \) means you move 1.5 units to the left of the origin.
• The y-coordinate \( -\frac{1}{3} \) means you move approximately 0.33 units down from there.
• Plot the second point on the graph.

By practicing plotting points, you will develop a visual understanding of how coordinates work together on a graph.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra and geometry. It provides a method to solve geometric problems through algebraic equations by using coordinate systems. Each point in coordinate geometry is represented by a pair of numbers, and these can be used to calculate distances, midpoints, and slopes.

A connection of points can form lines, curves, and other shapes, allowing complex geometric inferences to be made through simple algebraic processes. It’s crucial to understand that:

• A point is represented by its coordinates \((x, y)\) on the Cartesian plane.
• A line can be described by equations, such as linear equations.
• Properties like distance and section formula can be derived using coordinates.

Understanding coordinate geometry is essential since it lays the groundwork for more advanced mathematical fields. In the context of your exercise, calculating the slope or plotting a straight line requires the principles learned in coordinate geometry.

Linear Equations

Linear equations represent the simplest type of equation, where the highest power of the variable(s) is 1. They describe straight lines in coordinate geometry and are generally written in the form \( y = mx + c \), where:

• \( m \) is the slope of the line indicating its steepness and direction.
• \( c \) is the y-intercept, where the line crosses the y-axis.

The slope of a line is crucial as it reveals important information:

• If \( m > 0 \), the line rises from left to right.
• If \( m < 0 \), it falls from left to right.
• If \( m = 0 \), the line is horizontal.

Using the coordinates provided in your exercise, we calculated the slope \( m = -\frac{1}{7} \), which shows the line slightly falls as you move from left to right. This negative slope confirms that our line will trend downwards, showcasing a clear example of how linear equations operate within coordinate geometry.