Problem 27 - College Algebra with Corequisite Support

Question

For the following exercises, find the slope of the line that passes through the two given points. (8,-2) and (4,6)

Step-by-Step Solution

Answer

The slope of the line is -2.

Step 1: Determine the formula for the slope

The slope (m) of the line that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This formula calculates the change in the y-values divided by the change in the x-values.

Step 2: Assign the values to the formula

Identify the points: \((x_1, y_1) = (8, -2)\) and \((x_2, y_2) = (4, 6)\). Assign these values to the formula: \[ m = \frac{6 - (-2)}{4 - 8} \] This reflects subtracting the y-values and the x-values as per the formula.

Step 3: Calculate the numerator

Calculate the difference in the y-values:\[ 6 - (-2) = 6 + 2 = 8 \] This gives us the total change in the y-direction.

Step 4: Calculate the denominator

Calculate the difference in the x-values: \[ 4 - 8 = -4 \] This gives us the total change in the x-direction.

Step 5: Compute the slope

Substitute the calculated numerator and denominator back into the formula to find the slope: \[ m = \frac{8}{-4} = -2 \] This means the slope of the line is -2.

Key Concepts

Coordinate Geometry

In the land of mathematics, coordinate geometry acts as a bridge between algebra and geometry. It allows us to understand geometric shapes using algebra. By plotting points, lines, and curves on a plane, we can visualize and solve real-world problems.

Think of coordinate geometry as a map: the axes act like a grid reference, helping us locate points and lines. The horizontal line is called the x-axis, and the vertical line is the y-axis. The point where these two axes meet is the origin, marked as (0,0).

The beauty of coordinate geometry lies in its ability to precisely locate positions and determine relationships between different geometric entities. Understanding how lines are represented in this system is crucial since it's the foundation of many geometric interpretations, such as slope calculations and line equations.

Points

Points are the most basic concept in coordinate geometry. They are locations in a plane defined by a pair of numbers. These numbers, known as coordinates, are written in the form \((x, y)\).
For example, in the given exercise, the points (8, -2) and (4, 6) each have specific x and y values.

• The first number in the pair represents the x-coordinate, which tells us how far to move horizontally from the origin.
• The second number represents the y-coordinate, indicating how far to move vertically.

It is essential when solving problems like finding a slope, to first correctly identify and place each point on the coordinate plane. Understanding this helps us comprehend how far apart and in which direction the points lie from each other. This forms a foundational understanding of plotting linear equations, interpreting line graphs, and solving practical applications in various fields.

Linear Equations

Linear equations are pivotal in coordinate geometry. They represent lines and show the relationship between x and y coordinates. The slope, calculated as shown in the exercise, is an integral part of any linear equation.

Each line has its unique equation, typically written as \( y = mx + c \) where:

• \( m \) is the slope, indicating the steepness and direction of the line.
• \( c \) is the y-intercept, marking the point where the line crosses the y-axis.

The slope gives us insight into how much y will change with a unit change in x. In a practical sense, this can relate to real-world rates of change, such as speed, cost, or growth.

Understanding linear equations is not just about solving math problems but about grasping how two variables can interact in a predictable, linear manner. This understanding is crucial, as it helps us build models and predictions, deeply weaving into economics, physics, engineering, and numerous other fields.