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 1/3 and the line rises.

Step 1: Identify the given points

The given points are (-1, 3) and (2, 4). These can be written as \( (x_1, y_1) = (-1, 3) \) and \( (x_2, y_2) = (2, 4) \).

Step 2: Calculate the slope

Substitute the given values into the slope formula: \[ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} = \frac{{4 - 3}}{{2 - (-1)}} = \frac{1}{3} \]. Hence, the slope of the line passing through these points is 1/3.

Step 3: Determine the direction of the line

The slope 1/3 is positive, that means the line rises from left to right.

Key Concepts

Understanding Linear Equations

Linear equations form a fundamental part of algebra and are equations of the first order. They are remarkable for their simplicity and form the basis of many more complex calculations. A basic linear equation can always be written in the form:
\( y = mx + b \)
Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept, which is where the line crosses the y-axis.

• The variables \( x \) and \( y \) stand for any pair of values that make the equation true.
• In a graph, a linear equation is represented by a straight line.

Recognizing and understanding the structure and components of linear equations is crucial when analyzing lines in physics, economics, or everyday applications.

The Role of Coordinate Geometry

Coordinate geometry, also known as analytic geometry, bridges algebra and geometry using a coordinate plane to describe geometric shapes and analyze their properties. In this system:

• The plane is defined by two perpendicular axes: the x-axis (horizontal) and y-axis (vertical).
• Points on this plane are expressed as ordered pairs, such as \((x, y)\).

Using the given points \((-1, 3)\) and \((2, 4)\), we can graph these points on the plane to visualize the line they form.
Such visual representations help tremendously in solving problems involving distance, midpoint, and slope, thus making coordinate geometry an invaluable tool in both theoretical and practical applications.

Seizing the Concept of Positive Slope

A positive slope indicates a line that rises as it moves from left to right across the graph. With a positive slope, both the x and y values increase together, signifying a direct relation.

• The slope formula, \( m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \), is key to finding the steepness or tilt of this line.
• If \( m > 0 \), the slope is positive, as we calculated: \( \frac{1}{3} \).

This specific line representing a 1/3 slope ascends steadily as x increases.
Understanding a positive slope helps comprehend trends and relationships in real-world data, such as climbing profits or consumer demand with increasing price.

Exploring Rise and Run

When dealing with the slope of a line, the terms 'rise' and 'run' represent changes in the y-direction and x-direction, respectively. The slope is defined as
\( m = \frac{\text{rise}}{\text{run}} \).

• "Rise" is the vertical shift between two points on a line (change in y).
• "Run" is the horizontal shift (change in x).

For the points \((-1, 3)\) and \((2, 4)\), the rise is \(4 - 3 = 1\) while the run is \(2 - (-1) = 3\), giving a slope of \(\frac{1}{3}\).
Recognizing rise and run not only assists in calculating slopes but also in understanding the concept of slope in terms that are easy to visualize and apply, be it in hiking trails or engineering projects.