Problem 50 - Elementary Algebra Within Reach

Question

In Exercises 49-54, plot the points and find the slope (if possible) of the line that passes through the points. If not possible, state why. $$ (4,6),(8,-2) $$

Step-by-Step Solution

Answer

The slope of the line that passes through the points (4,6) and (8,-2) is -2.

Step 1: Plot the given points

Draw a Cartesian plane and plot the points (4,6) and (8,-2). Write down the coordinates for easy reference.

Step 2: Calculating the slope

To determine the slope (m) of a line through two given points (x1,y1) and (x2,y2), we use the formula \(m = (y2-y1) / (x2-x1)\). Here, (x1,y1) = (4,6) and (x2,y2) = (8,-2). Substituting these values into the formula, we get \(m = (-2 - 6) / (8 - 4)\).

Step 3: Simplifying the expression

Simplify the expression we got in the previous step to calculate the slope \(m\). So, \(m = -8 / 4\).

Key Concepts

Plotting Points

Plotting points is the first step in many coordinate geometry problems. It involves marking specific locations on the Cartesian plane using ordered pairs, known as coordinates.
Each point is defined by an

• x-coordinate (horizontal position)
• y-coordinate (vertical position)

For the points (4,6) and (8,-2), the number 4 and 8 represent the horizontal positions, and 6 and -2 indicate their vertical positions on the plane.
When plotting these points:

• Start from the origin (0,0), the center of the plane.
• Move horizontally to the x-coordinate.
• Move vertically to the y-coordinate.

This method helps visualize where each point is located and is essential for understanding the relationship between them.

Cartesian Plane

The Cartesian plane is a two-dimensional surface defined by a horizontal axis (x-axis) and a vertical axis (y-axis).
It is used in coordinate geometry to plot points, lines, and shapes. The point where the x-axis and y-axis intersect is called the origin, represented by (0,0).
The plane is divided into four quadrants:

• Quadrant I: Positive x and y values
• Quadrant II: Negative x and positive y values
• Quadrant III: Negative x and y values
• Quadrant IV: Positive x and negative y values

Understanding the quadrants helps determine the position of points, such as (4,6) in Quadrant I and (8,-2) in Quadrant IV, making it easier to analyze line slopes and geometrical shapes.

Calculating Slope

Calculating the slope of a line is crucial for understanding its steepness and direction. The slope, denoted by the symbol \(m\), is determined by examining two points on the line.
The formula for calculating the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
For our example, the points (4,6) and (8,-2) are used:

• \(x_1 = 4\), \(y_1 = 6\)
• \(x_2 = 8\), \(y_2 = -2\)

By substituting the values into the formula, we calculate:\[m = \frac{-2 - 6}{8 - 4} = \frac{-8}{4}\]The result is \(m = -2\), indicating the line slopes downward to the right.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, bridges algebra and geometry using a coordinate system. It allows the analysis and representation of geometric shapes through algebraic equations.
This branch of mathematics is essential in finding distances, midpoints, and slopes of lines in the plane.

• Points indicate specific locations.
• Lines can be defined using slope and points.
• Shapes are described algebraically.

In our example, coordinate geometry techniques were employed to locate points on the Cartesian plane and determine the slope of a line through these points. Understanding these concepts enhances problem-solving skills and develops a deeper comprehension of geometry and algebra interactions.