Problem 26 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 23-28, find the slope of the line containing the given two points. \((2,-4)\) and \((0,-6)\)

Step-by-Step Solution

Answer

The slope of the line is 1.

Step 1: Identify the two points

First, identify the two points given in the problem: Point 1 is \(2, -4\) and Point 2 is \(0, -6\). These points are represented in the format \(x_1, y_1\) and \(x_2, y_2\), respectively.

Step 2: Use the slope formula

The formula for finding the slope \(m\) of a line given two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \( m = \frac{y_2 - y_1}{x_2 - x_1}\). This formula represents the change in the y-coordinates divided by the change in the x-coordinates.

Step 3: Substitute values into the formula

Substitute the values from the points into the slope formula: \( m = \frac{-6 - (-4)}{0 - 2} = \frac{-6 + 4}{0 - 2}\).

Step 4: Simplify the expression

Now, simplify the expression: \(m = \frac{-2}{-2}\).

Step 5: Calculate the slope

Finish simplifying the fraction: \(m = 1\). Therefore, the slope of the line passing through the points \(2, -4\) and \(0, -6\) is 1.

Key Concepts

Coordinate Geometry

Coordinate geometry is the study of geometry using a coordinate system. It combines algebra and geometry using coordinates to explore geometric properties. In two dimensions, we often use the Cartesian coordinate system, which consists of two axes: the x-axis (horizontal) and the y-axis (vertical). Each point in this system is represented as an ordered pair \(x, y\).

Points allow us to locate objects in this space, making it easier to handle geometric problems numerically. This system helps us explore shapes, positions, and lines on a plane. By plotting points, like \((2, -4)\) and \(0, -6)\), we start forming the visual representation of lines and curves.

• Points have coordinates \( (x, y) \) format.
• The x-coordinate indicates horizontal movement.
• The y-coordinate indicates vertical movement.

Using coordinate geometry, solving geometric problems becomes more intuitive and precise.

Slope Formula

The slope formula is a crucial part of understanding linear equations and lines in coordinate geometry. The slope of a line, often denoted as \( m \), measures the steepness or incline of that line. It's a ratio indicating how much the line rises or falls vertically for every unit it moves horizontally.

The formula to find the slope given two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This formula shows the change in y-coordinates (vertical change) divided by the change in x-coordinates (horizontal change).

Example: Calculating the Slope

Using the points from our exercise, \((2, -4)\) and \(0, -6)\), we substitute into the formula:

• Subtract the y-coordinates: \(-6 - (-4) = -6 + 4 = -2\)
• Subtract the x-coordinates: \((0 - 2) = -2\)
• Divide the differences: \(\frac{-2}{-2} = 1\)

This slope indicates a line that rises one unit for every unit it moves to the right.

Linear Equations

Linear equations represent straight lines on a graph. They are fundamental in algebra and coordinate geometry. A linear equation can be written in different forms, with the slope-intercept form being one of the most common:

• \( y = mx + b \)

Here, \( m \) is the slope, and \( b \) is the y-intercept (where the line crosses the y-axis).

The slope \( m \) reflects the rate of change, while \( b \) shows the starting value when \( x = 0 \). Understanding this helps in predicting how changes in one variable affect another. Through solving linear equations, one learns how to compute the output for given inputs to sketch lines that embody algebraic expressions.

Connecting Points and Lines

Given two points, we can determine the equation of the line that connects them. By calculating the slope and recognizing the intercept, we describe the line's path on the graph accurately.

Mathematics Education

In mathematics education, building conceptual understanding of topics like slope and linear equations is much more than procedural knowledge. Knowing how to compute the slope is important, but understanding its meaning is crucial in real-world applications. This requires emphasizing:

• Why the slope is a ratio of changes.
• How it describes direction and steepness.
• Relating concepts to tangible scenarios, like roads or ramps.

These concepts are the building blocks for advanced topics in mathematics and various STEM fields.

Encouraging students to visualize problems through graphing reinforces their understanding. Practical exercises, like finding the slope between two points, make the principles of coordinate geometry more relatable and easier to grasp. Interactive teaching methods, which blend technology and traditional techniques, help students gain confidence and clarity in solving mathematical problems.