Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 23-28, find the slope of the line containing the given two points. \((3,0)\) and \((0,5)\)

Step-by-Step Solution

Answer

The slope of the line is \(-\frac{5}{3}\).

Step 1: Understanding Slope Formula

The slope of a line containing two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). We will use this formula to find the slope of the line containing the points (3,0) and (0,5).

Step 2: Substituting Values

Substitute the given points into the slope formula. Here, \( x_1 = 3 \), \( y_1 = 0 \), \( x_2 = 0 \), and \( y_2 = 5 \). The expression becomes: \( m = \frac{5 - 0}{0 - 3} \).

Step 3: Calculating the Difference

Calculate the differences \( y_2 - y_1 \) and \( x_2 - x_1 \). This gives us \( 5 - 0 = 5 \) and \( 0 - 3 = -3 \). Substitute these values back into the formula to get \( m = \frac{5}{-3} \).

Step 4: Simplifying the Slope

Simplify the fraction. \( m = \frac{5}{-3} \) simplifies to \( m = -\frac{5}{3} \). This is the slope of the line passing through the points (3,0) and (0,5).

Key Concepts

Coordinate Geometry

Coordinate geometry, often referred to as analytic geometry, is a branch of mathematics that connects geometry and algebra through the use of a coordinate system. In this system, points are defined by a pair of numerical coordinates. These coordinates represent the distance from two perpendicular lines, known as the axes.
This approach allows us to describe geometric shapes and lines algebraically, leading to a better understanding of their properties and relationships.

• Each point in the plane is represented by a pair of numbers \( (x, y) \), where \( x \) indicates the horizontal position and \( y \) indicates the vertical position.
• Lines can be studied using linear equations, such as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

This fusion of algebra and geometry provides tools to solve problems involving shapes, positions, and other spatial concepts. Coordinate geometry is foundational for calculus and other advanced mathematical areas. Understanding this concept enhances your problem-solving skills and your ability to visualize problems geometrically.

Slope Formula

The slope of a line is a fundamental concept in coordinate geometry that measures the steepness of a line. Specifically, it defines how much a line inclines or declines as it moves along the x-axis. The slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \) calculates the slope of a line passing through any two points \( (x_1, y_1) \) and \( (x_2, y_2) \).
To find the slope:

• Identify two points on the line.
• Subtract the y-coordinates (rise) and x-coordinates (run).
• Divide the difference of the y-coordinates by the difference of the x-coordinates.
• The result is the slope, often represented as 'm.'

For example, using points (3,0) and (0,5), the slope is calculated as follows:
\( m = \frac{5 - 0}{0 - 3} = \frac{5}{-3} = -\frac{5}{3} \)
The negative slope indicates the line falls as it moves from left to right.

Points on a Line

Points on a line refer to specific locations along that line in a coordinate plane. Each point is characterized by its coordinate pair \( (x, y) \). These points are essential in plotting the line on a graph and are used to determine the line's slope.
Points can tell us much about the line, including its direction, steepness, and position:

• Two distinct points are sufficient to define a straight line in a plane.
• By substituting point coordinates into the slope formula, we can determine the line's slope.
• The line equation \( y = mx + b \) can be determined using these points, where 'b' is the y-intercept.

Calculating the slope using any two points on the line helps us understand how the line traverses across the coordinate plane. The specific example of points (3,0) and (0,5) allows us to calculate a slope of -5/3, indicating every three units the line moves horizontally to the left, it falls five units vertically.