Problem 41 - Precalculus with Limits

Question

In Exercises 41-50, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.) \( (2, 1) \), \( m = 0 \)

Step-by-Step Solution

Answer

The three additional points through which the line passes are: (0,1), (1,1), and (3,1).

Step 1: Realize that the slope of the line is zero

From the problem, the line has a slope \( m = 0 \). This means the line is horizontal.

Step 2: Apply the Slope-Intercept Form of the Equation

According to the slope-intercept form \( y = mx + c \), where m is the slope and c is the y-intercept, since \( m = 0 \), the equation is \( y = c \). Given the point \( (2, 1) \), we calculate the y-coordinate which remains constant throughout since m=0 resulting into \( y = 1 \).

Step 3: Find three additional points

Now, let's find three points on the line. As the y-coordinate is constant and equal to 1, we can choose any x-coordinate. A good choice could be \( x = 0 \), \( x = 1 \), and \( x = 3 \) (avoiding the already given point \( x = 2 \)). Hence, the three additional points are \( (0,1) \), \( (1,1) \), and \( (3,1) \).

Key Concepts

Equation of a Line

The concept of the "Equation of a Line" forms the foundation of understanding relationships between variables in coordinate geometry. An equation of a line essentially describes how the x-coordinates and y-coordinates relate to each other on a 2D plane.

In general, the equation of a line can be expressed in several forms, but one of the simplest is the **slope-intercept form**. To establish this equation, you need to identify a specific point on the line and the line's slope. With this information, you can precisely chart out the line's path across a graph.

For any given line, no matter how it is oriented, an equation serves as its mathematical representation, enabling predictions and calculations of unseen points along that line.

Slope-Intercept Form

The slope-intercept form of a line is a convenient and intuitive way to express linear equations. It is given by the formula: \[ y = mx + b \]where:

• \( y \) is the dependent variable (the output value on the y-axis),
• \( m \) is the slope of the line, and
• \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

Understanding this concept allows you to quickly visualize how changes in x affect y.

In cases of horizontal lines like the one described by \( m = 0 \), the slope-intercept form simplifies significantly. Instead of the slope influencing the relationship between x and y, y remains constant, reflecting directly at the y-intercept. So, the equation, in this case, becomes:\[ y = b \]

For our specific exercise, given the point (2,1), the equation of the line becomes \( y = 1 \), demonstrating a flat line where the y-value does not change regardless of the x-value.

Finding Points on a Line

Locating additional points on a line when given a slope and one point can seem challenging initially. However, with a bit of practice, it becomes straightforward.

For horizontal lines, defined by a slope of zero, every point on the line shares the same y-value as the given point. The only thing altering is the x-coordinate, which can be freely chosen unless a particular restriction applies.

Imagine you have the line equation \( y = 1 \). To find more points, you simply select different values for \( x \). From our exercise, choosing values like \( x = 0 \), \( x = 1 \), and \( x = 3 \) gives new points:

• (0, 1)
• (1, 1)
• (3, 1)

Each of these points satisfies the equation because the y-coordinate is consistently 1, aligned with the determined line equation. This approach allows flexibility and ease in plotting out points along any horizontal line.