Problem 29 - Precalculus with Limits

Question

In Exercises 29-40, plot the points and find the slope of the line passing through the pair of points. \( (0, 9) \), \( (6, 0) \)

Step-by-Step Solution

Answer

The slope of the line passing through the points (0, 9) and (6, 0) is -1.5.

Step 1: Identify the Given Points

The given points are (0,9) and (6,0). The first coordinate in each pair is the x-coordinate, and the second is the y-coordinate. So, the points are located at (x1,y1) = (0,9) and (x2,y2) = (6,0).

Step 2: Plot the Points

The points (0,9) and (6,0) are plotted on a graph. The first point (0,9) is located at the y-intercept (where the line crosses the y-axis). The second point (6,0) is located at the x-axis where the line crosses the x-axis.

Step 3: Calculate the Slope

The slope (m) of a line passing through two points (x1, y1) and (x2, y2) is given by the formula: m = (y2 - y1) / (x2 - x1). Substituting the given points into this formula, the slope m is calculated as m = (0 - 9) / (6 - 0) = -9 / 6 = -1.5.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional space where we can represent points using pairs of numbers. It is divided into four quadrants by a horizontal line, called the x-axis, and a vertical line, called the y-axis. These axes intersect at the origin, which is the point (0,0). The position of any point on this plane is determined by its x-coordinate and y-coordinate.

• The x-coordinate tells us how far to move left or right from the origin.
• The y-coordinate tells us how far to move up or down from the origin.

By understanding the structure of the coordinate plane, we can easily locate and plot points to visualize mathematical concepts like lines and shapes. Navigating through this grid allows us to better understand the relationships between different points and lines.

Plotting Points

Plotting points on a coordinate plane is like placing dots on a map. To plot a point, you need to identify its coordinates \(x, y\). The x-coordinate tells you how far to move horizontally, and the y-coordinate shows how far to move vertically.

• For instance, to plot the point (0,9): Move 0 units along the x-axis (stay at the origin), and then move 9 units up.
• For point (6,0): Move 6 units along the x-axis, and because the y-coordinate is 0, stay on the x-axis.

Once points are plotted, they can be connected to form shapes or lines. This helps in visualizing equations and understanding the nature of functions or mathematical relationships.

Linear Equations

A linear equation represents a line on the coordinate plane and generally has the form \ y = mx + b \, where:

• \( y \) is the value on the y-axis, \( x \) is the value on the x-axis,
• \( m \) represents the slope of the line, which shows how steep the line is, and \( b \) is the y-intercept (the point where the line crosses the y-axis).

In the given exercise, we calculated the slope, \( m = -1.5 \), using the formula for slope calculation: \ m = \frac{y_2 - y_1}{x_2 - x_1} \. This measures the change in y per unit change in x. The negative slope indicates the line slants downward from left to right. Understanding linear equations allows us to predict values and understand the relationships between variables in various contexts.