Problem 30 - Precalculus with Limits

Question

In Exercises 29-40, plot the points and find the slope of the line passing through the pair of points. \( (12, 0) \), \( (0, -8) \)

Step-by-Step Solution

Answer

The slope of the line passing through the points \( (12, 0) \) and \( (0, -8) \) is \( \frac{{2}}{{3}} \).

Step 1: Plot the Points

The given points are \( (12, 0) \) and \( (0, -8) \). Plot these points on a graph.

Step 2: Use the Slope Formula

For the points \( (12, 0) \) and \( (0, -8) \), identify the \(x\) and \(y\) coordinates as \(x1, y1\) and \(x2, y2\). \(x1 = 12\), \(y1 = 0\), \(x2 = 0\), and \(y2 = -8\). Substituting these into the slope formula, we get \(m = \frac{{-8 - 0}}{{0 - 12}}\).

Step 3: Calculation

Solving the expression, we get: \(m = \frac{{-8}}{{-12}} = \frac{{2}}{{3}}\).

Key Concepts

Plotting Points on a Graph

Plotting points is a foundational skill in coordinate geometry. It's about marking exact locations on a graph that correspond to a given pair of coordinates. In a two-dimensional coordinate plane, each point is represented by an ordered pair, typically in the form \( (x, y) \). This pair tells you where to place the point on the graph:

• The first number, \( x \), determines the horizontal position. You move right if \( x \) is positive, and left if \( x \) is negative.
• The second number, \( y \), determines the vertical position. You move up for positive \( y \) values and down for negative ones.

For instance, given the point \( (12, 0) \), you'd start at the origin and move 12 units to the right along the \( x \)-axis. Since \( y \) is zero, you stay on the axis. For the point \( (0, -8) \), you'd stay on the \( y \)-axis and move 8 units downward. It is essential to plot points accurately, as accurate representation impacts further calculations.

Understanding the Slope Formula

The slope of a line is a measure of its steepness and direction. To find the slope between two points, you can use the slope formula, \( m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \). Here's how each part of the formula works:

• \( y_2 - y_1 \) is the change in the vertical position or the rise.
• \( x_2 - x_1 \) is the change in the horizontal position or the run.
• \( m \) is the symbol used to denote the slope.

For example, with points \( (12, 0) \) and \( (0, -8) \), assigning \( x_1 = 12 \), \( y_1 = 0 \), \( x_2 = 0 \), and \( y_2 = -8 \), you substitute these into the formula, obtaining: \( m = \frac{{-8 - 0}}{{0 - 12}} \), which simplifies to \( m = \frac{{-8}}{{-12}} \),resulting in the slope \( \frac{{2}}{{3}} \). The positive outcome suggests that the line inclines upward from left to right.

Applying Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This way of studying geometry allows you to solve problems involving distances, slopes, and patterns by using coordinates and formulas. The essentials of coordinate geometry include:

• Coordinate System: Defined by an \( x \)-axis and a \( y \)-axis, it allows you to describe the location of points uniquely.
• Points: Each point on the plane is described by an ordered pair \( (x, y) \).
• Lines: Using the slope formula, you can describe the incline of a line between any two points.

In the context of our exercise, you use coordinate geometry to visualize and calculate. By plotting the given points on the plane and using the slope formula, you define the linear relationship between them. This interaction of algebra and geometry is what makes coordinate geometry such a powerful tool in mathematics.