Problem 49 - 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.) \( (7, -2) \), \( m = \frac{1}{2} \)

Step-by-Step Solution

Answer

Three additional points on the line are (9, -1), (11, 0), and (13, 1).

Step 1: Understanding the slope

The slope, denoted by 'm', is the amount that y increases/decreases for each increase of 1 in x. Here, our slope \( m = \frac{1}{2} \) means that for every increase of 2 in x, y increases by 1 (since 1 divided by 2 is \( \frac{1}{2} \)).

Step 2: Calculating new points

We can start from the given point (7, -2) and follow the slope to discover additional points. Starting from an initial point, an increase in 'x' by a certain value and a corresponding increase/decrease in 'y' using the slope yields new points on the line. Following this, for an increase in x by 2 units, there is an increase in y by 1 unit.

Step 3: Applying to our data

Starting from the given point (7, -2), if we increase x by 2, we get 7 + 2 = 9 and the corresponding y = -2 + 1 = -1. Therefore, the new point is (9, -1). Similarly, if we increase x by another 2 units, we get x = 9 + 2 = 11 and y = -1 + 1 = 0. Thus, the second new point is (11, 0). Again, increasing x by 2 more units, we get x = 11 + 2 = 13 and y = 0 + 1 = 1. Thus, the third new point is (13, 1).

Key Concepts

Slope of a Line

The slope of a line is a crucial concept in algebra and geometry, revealing both the direction and steepness of the line. When we represent a line by its equation, the slope is commonly symbolized by the letter 'm'. In a graph, it is shown as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. A positive slope means the line is ascending, while a negative slope indicates it descends as we move from left to right.

For example, the slope of \( m = \frac{1}{2} \) indicates that for every two units we move to the right (run), the line goes up by one unit (rise). to calculate new coordinate points that lie on the same line, you simply apply the slope incrementally. If a slope is presented as a fraction, like \( m = \frac{a}{b} \) , it means for every 'b' units you move to the right, the line moves 'a' units vertically.

Coordinate Points

Coordinate points are the sets of numbers that represent positions on a graph. Typically denoted as (x, y), these coordinates show where a point is located in relation to the horizontal (x) and vertical (y) axes. When determining the location of points on a line, we start with an original point—such as (7, -2) in our example—and use the line's slope to find additional points.

For this, you'll often involve adding or subtracting from the original x and y based on the slope. In our case, since the slope is \( m = \frac{1}{2} \) for every 2 units increased in x, the y-value increases by 1. This method allows plotting multiple points to draw the line graphically, ensuring that they're all aligned as per the slope's indication.

Linear Equations

Linear equations represent lines on the coordinate plane and can be written in various forms, including slope-intercept form, point-slope form, and standard form. The most common form used in elementary algebra is the slope-intercept form, denoted as y = mx + b, where 'm' is the slope and 'b' is the y-intercept, the point where the line crosses the y-axis.

Understanding linear equations is foundational for solving algebraic problems, as it represents the relationship between two variables. If you know the slope and at least one point on the line, you can write the equation of the line. Connecting linear equations with slope and coordinate points completes the picture of how algebraic expressions model real-life situations and how changing one variable affects another within a system.