Problem 19 - Algebra for College Students

Question

Find the slope of the line that contains each of the following pairs of points. \((2,6),(5,1)\)

Step-by-Step Solution

Answer

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

Step 1: Identify the coordinates

Given points are \(2,6\) and \(5,1\). Label the coordinates as follows: \( (x_1, y_1) = (2, 6) \) and \( (x_2, y_2) = (5, 1) \).

Step 2: Use the slope formula

The formula to find the slope of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].

Step 3: Substitute the coordinates

Substitute \((x_1, y_1) = (2, 6)\) and \((x_2, y_2) = (5, 1)\) into the slope formula: \[ m = \frac{1 - 6}{5 - 2} \].

Step 4: Simplify the expression

Calculate the values in the numerator and the denominator: \[ m = \frac{-5}{3} \]. Hence, the slope of the line is \(-\frac{5}{3}\).

Key Concepts

slope formula

Finding the slope of a line is a fundamental concept in coordinate geometry. The slope formula helps us determine the steepness and direction of a line. When you have two points, \( (x_1, y_1) \) and \( (x_2, y_2) \), the slope \( m \) is found using the following formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \].
This formula calculates the change in y-coordinates divided by the change in x-coordinates, often referred to as 'rise over run'.
It's crucial to remember this:

• Nominate the coordinates strictly; decide which point is \( (x_1,y_1) \) and which is \( (x_2,y_2) \).
• Substitute the values correctly into the formula.
• Subtract \( y_1 \) from \( y_2 \) and \( x_1 \) from \( x_2 \).

Practicing this consistent method will help you find the slope easily!

coordinate geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system.
It allows the geometric shapes to be analyzed and understood by reducing them to algebraic equations.
To find the slope of a line, you need to address these key concepts:

• Coordinates: A pair that shows a position on the plane. Example: \( (2, 6) \).
• Axes: The x-axis (horizontal line) and y-axis (vertical line) intersect at the origin \( (0,0) \).
• The Cartesian Plane: The plane containing these axes is divided into four quadrants that help in locating points.

Using the Cartesian coordinate system simplifies solving problems like calculating the slope or finding distances between points.

linear equations

Linear equations represent straight lines in the coordinate plane. An essential use of the slope is to describe these lines.
Any linear equation can be represented in the form \[ y = mx + c \].
Here,

• \( m \): represents the slope of the line,
• \( x \) and \( y \): are the variables representing coordinates,
• \( c \): is the y-intercept, the point where the line crosses the y-axis.

Finding the slope enables us to understand the direction and steepness of the line.
Remember:
A positive slope means the line rises, and a negative slope means it falls as you move from left to right. This concept of linear equations is fundamental in learning coordinate geometry and studying algebra.