Problem 2 - Introductory Algebra for College Students

Question

In Exercises \(1-10,\) 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. \((2,1)\) and \((3,4)\)

Step-by-Step Solution

Answer

The slope of the line passing through the points \((2,1)\) and \((3,4)\) is 3, indicating that the line rises.

Step 1: Identify the Coordinates

First, identify the coordinates of the two given points. Here, \((x_1, y_1) = (2,1)\) and \((x_2, y_2) = (3,4)\).

Step 2: Apply the Slope Formula

Next, substitute these coordinates into the formula for the slope, that is, Slope = \(\frac{y_2-y_1}{x_2-x_1}\). Hence, Slope = \(\frac{4-1}{3-2} = \frac{3}{1} = 3\).

Step 3: Analyze the Slope

With the slope being positive, it can be concluded that the line rises. A positive slope indicates a rising line, a negative slope indicates a falling line, a slope of zero indicates a horizontal line, and an undefined slope (when the denominator in the slope formula is zero) indicates a vertical line.

Key Concepts

Linear Equations

Linear equations represent the relationship between two variables in a straight line when graphed on a coordinate plane. They are often written in the slope-intercept form, which is \[ y = mx + b \] \ where \( m \) represents the slope of the line, and \( b \) is the y-intercept, the point where the line crosses the y-axis.

• The slope \( m \) tells us how steep the line is and in what direction it slants.
• The equation of a line helps to determine the relationship between the variables for any given value.
• Linear equations can also be rearranged into standard form: \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants.

Understanding linear equations is crucial, as they form the foundation for studying more advanced topics in algebra and calculus. They are not only used in mathematics but also have practical applications in fields like economics, physics, and engineering.

Coordinate Geometry

Coordinate geometry, also known as analytical geometry, connects the algebraic equations with geometric representations. It helps us to visualise geometrical shapes and forms by using a coordinate plane.
In coordinate geometry:

• Points are represented by pairs of numbers \((x, y)\), called coordinates.
• The horizontal axis is known as the x-axis, while the vertical axis is the y-axis.
• This provides a framework to study and make sense of shapes, distances, and angles in a flat space.

To find the distance between two points, we use the distance formula: \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Coordinate geometry forms the backbone of various mathematical concepts and is instrumental in solving real-world problems involving shapes and space.

Positive and Negative Slope

The slope of a line is a measure that shows how steep the line is and the direction in which it moves. It is calculated using the change in y-coordinates (vertical change) divided by the change in x-coordinates (horizontal change).

A **positive slope** means that as you move from left to right on the graph, the line goes upwards. This indicates a rising line.

• If you visualize the slope as a hill, a positive slope would be an uphill walk.

Conversely, a **negative slope** tells us that the line falls as you move from left to right.

• Here, it's akin to walking downhill.

A **zero slope** implies a horizontal line. It means there is no vertical change as the x-values increase.

Meanwhile, an **undefined slope** corresponds to a vertical line where the x-values remain constant no matter the y-value. Understanding slopes is essential for graphing linear functions accurately and for predicting how changes in variables will impact the graph of the line.