Problem 4 - College Algebra

Question

Find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. \((-1,3)\) and \((2,4)\)

Step-by-Step Solution

Answer

The slope of the line passing through the points (-1,3) and (2,4) is \( \frac{1}{3} \). The line rises.

Step 1: Identify the coordinates of the points

The coordinates of the two points are: Point 1 = (-1,3) and Point 2 = (2,4). So, \( x_{1} = -1, y_{1} = 3, x_{2} = 2, y_{2} = 4 \)

Step 2: Apply the formula to find the slope

Using the formula to calculate the slope, represented as 'm', we get: \( m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{4 - 3}{2 - (-1)} = \frac{1}{3} \)

Step 3: Determine whether the line rises, falls, is horizontal, or is vertical

Since the slope \( m = \frac{1}{3} \) is positive, it is concluded that the line rises.

Key Concepts

Coordinates

Coordinates are essential in geometry as they represent a point's position in space. A coordinate pair, such as (-1, 3) and (2, 4), consists of two numbers. The first number of the pair represents the horizontal position on the x-axis, and the second number represents the vertical position on the y-axis. When dealing with coordinates, remember:

• Each point has an X-coordinate and a Y-coordinate.
• X-coordinates indicate horizontal placement.
• Y-coordinates indicate vertical placement.

Coordinates are fundamental for understanding graphs and equations since they allow us to locate points on a plane. Without this system of numerical addresses, it would be impossible to accurately represent and connect points within any geometric space. Understanding how coordinates are structured and used is key to grasping more complex mathematical concepts and applications.

Slope Formula

The slope of a line tells us how steep the line is, or in simple terms, the rate at which one variable changes relative to another in a linear equation. This is often represented by the letter \( m \). To calculate the slope of a line passing through two points, there is a simple formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here:

• \( y_2 \) and \( y_1 \) are the y-coordinates of the two points.
• \( x_2 \) and \( x_1 \) are the x-coordinates of the two points.

The formula works by determining the difference between the y-coordinates and dividing it by the difference between the x-coordinates, essentially giving the rise over the run.For example, using the points (-1, 3) and (2, 4):

• The difference in y-coordinates is 4 - 3 = 1 (the rise).
• The difference in x-coordinates is 2 - (-1) = 3 (the run).
• So, \( m = \frac{1}{3} \).

The slope \( \frac{1}{3} \) means that for every 3 units you move horizontally, you move 1 unit up vertically, indicating that the line rises as it moves from left to right. This calculation not only helps in determining the direction of the line but also in understanding relationships between variables.

Linear Equations

Linear equations make up a large part of algebra and geometry by representing straight lines in a coordinate system. A basic form of a linear equation is: \[ y = mx + b \] Here:

• \( y \) is the dependent variable (often output),
• \( x \) is the independent variable (input),
• \( m \) is the slope of the line,
• \( b \) is the y-intercept, where the line crosses the y-axis.

Linear equations are used to model relationships where the change in one variable consistently affects the other, such as speed and time, or price and quantity. When graphing a linear equation, you start at the y-intercept (point \( b \)) and use the slope \( m \) to calculate the rise over run to find other points on the line. Understanding how to set up and interpret these equations is essential for solving and graphing problems within algebra, physics, economics, and more. Linear equations are powerful because they can represent simple relationships clearly and are a gateway to more complex functions and models used in various fields of study.