Problem 25 - Precalculus with Limits

Question

In Exercises 17-28, find the slope and \(y\)-intercept (if possible) of the equation of the line. Sketch the line. \( y - 3 = 0 \)

Step-by-Step Solution

Answer

The slope of the line \( y - 3 = 0 \) is 0 and the y-intercept is 3. The graph of the line is a horizontal line crossing the y-axis at y = 3.

Step 1: Identify the equation's form

The given equation \( y - 3 = 0 \) can be rewritten in the slope-intercept form as \( y = 0x + 3 \).

Step 2: Identify the slope

The coefficient of x in the equation (which is 0 in this case) is the slope of the line. So, the slope (m) is 0.

Step 3: Identify the y-intercept

The constant term in the equation is the y-intercept (b). So, the y-intercept is 3.

Step 4: Sketch the line

A line with a slope of 0 is a horizontal line. So, the line y = 3 is a horizontal line that crosses the y-axis at 3.

Key Concepts

Slope of a Line

The slope of a line is a fundamental concept in algebra that describes how steep a line is. You can think of the slope as the "tilt" of the line. It tells us how much the y-value of a point on the line changes as we move one unit along the x-axis.

In mathematical terms, the slope (often represented by the letter "m") is calculated as the ratio of the change in y to the change in x. The formula to find the slope between two points \(x_1, y_1\) and \(x_2, y_2\) on a line is:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

In the exercise example, the line equation is \(y = 0x + 3\), indicating a slope of 0. This means the line does not "tilt" or "rise" and is perfectly horizontal.

Y-Intercept

The y-intercept of a line refers to the point where the line crosses the y-axis. It is a vital concept in understanding the position of the line on a graph. The y-intercept is represented as a constant in the slope-intercept form of a line, \(y = mx + b\), where \(b\) is the y-intercept.

It's important because it gives us a specific location of the line relative to the vertical y-axis. In our original exercise, the equation \(y = 0x + 3\) shows us that the y-intercept is 3, meaning the line crosses the y-axis at the point (0, 3).

• The y-intercept is always written as a singular value, as it's a point on the y-axis where \(x = 0\).
• Knowing the y-intercept makes it easier to visualize or sketch the line on a graph.

Horizontal Line

A horizontal line is a simple yet critical concept in geometry and algebra. It is defined by having a constant y-value for all x-values and, therefore, a slope of 0.

This type of line runs parallel to the x-axis and is neither rising nor falling as it moves along the axes. In our example, the line \(y = 3\) is a horizontal line. Every point on this line has a y-value of 3, making the line perfectly flat.

• Horizontal lines have no incline, so their slope is always 0.
• They are represented as \(y = c\), where \(c\) is a constant, identifying the y-value everywhere on the line.

Understanding horizontal lines is crucial because they represent situations with no change in the vertical (y) direction.

Equation of a Line

The equation of a line is a way of defining lines using mathematical expressions. It provides a clear and concise way to describe the line's direction and position in a coordinate plane. The most common form is the slope-intercept form: \(y = mx + b\).

In this format, \(m\) denotes the slope of the line and \(b\) represents the y-intercept. This makes it easy to quickly identify how steep the line is (its slope) and where it crosses the y-axis (the y-intercept).

For a horizontal line, as shown in the example \(y = 3\), the equation is slightly different since there is no x-term. This implies that the slope is zero, and the line remains constant at y = 3 regardless of any x-values.

• Slope-intercept form helps in sketching the line by providing quick reference points.
• It provides a direct insight into the properties of the line, such as steepness and starting point.

Being familiar with the equation of a line allows for quick graphing and understanding of linear relationships.