Problem 17 - College Algebra and Calculus: An Applied Approach

Question

Use the point on the line and the slope of the line to find three additional points through which the line passes. \(\begin{array}{ll}\text { Point } & \text { Slope }\end{array}\) \((5,-2) \quad m=0\)

Step-by-Step Solution

Answer

The three additional points through which the line passes are (-3, -2), (0, -2) and (8, -2).

Step 1: Understanding the Line Equation

For any line, the line equation is expressed as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In the given problem, the slope, \(m\), is given as zero. So, the equation becomes \(y = 0*x + b\), which simplifies to \(y = b\).

Step 2: Substitute the Known Point

The y-intercept (b) can be found by substituting the known point into the line equation. The point given is (5, -2). Since y-coordinate is -2, it's clear that b=-2. So the equation of line becomes \( y= -2 \).

Step 3: Finding Three Additional Points

Since the line equation is \(y=-2\) and the y-coordinate remains the same (as it's a horizontal line), we can select any three x-values and each time the y-value will be -2. Let's select x = -3, x = 0 and x = 8. Hence, the three additional points through which the line passes are (-3, -2), (0, -2) and (8, -2).

Key Concepts

Linear Equations

A linear equation is the foundation of algebra and it represents a straight line on a graph. Every linear equation can be written in the form of \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) is the y-intercept, which is the point where the line crosses the y-axis. These equations come in handy when you're dealing with constant rates of change. A distinctive feature of linear equations is that their graph will always be a straight line. The simplicity of these equations makes them very useful for modeling real-world situations that involve a straight-line relationship between two variables.

For instance, imagine you're starting a business and you want to predict your total costs based on the number of units produced — that's often a linear relationship, at least within a certain range. In our textbook exercise, we have a line with a slope (\(m\)) of zero, indicating that the change in \(y\) with respect to \(x\) is zero, therefore, it’s a horizontal line.

Slope of a Line

The slope is a measure of the steepness and the direction of a line. If you think of a line graph as a hill, the slope tells you how steep the hill is. It's calculated by taking the vertical change (rise) and dividing it by the horizontal change (run) between any two points on a line. In the form \(m = \frac{rise}{run}\), we call it \(m\) for short. The slope can be positive, negative, zero, or undefined.

• If the slope is positive (\(m > 0\)), the line rises as it moves from left to right.
• If it's negative (\(m < 0\)), the line falls from left to right.
• A slope of zero (\(m = 0\)), like we see in our exercise, means the line is perfectly horizontal and there is no rise (or fall) as you move along the line.
• An undefined slope (where run equals zero) is a vertical line, and it means the line goes straight up and down.

Now, why is this concept so important? Because the slope allows us to predict and describe the behavior of the line. Most importantly, in a scenario where the slope is zero, like in our given problem, knowing this fact simplifies finding additional points since \(y\) remains constant.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis on a graph. It's the value of \(y\) when \(x = 0\). In the linear equation \(y = mx + b\), the y-intercept is represented by \(b\). This concept helps us anchor our line on the graph; once you know the y-intercept and the slope, you can graph the entire line. A line can have only one y-intercept, but it can intersect the x-axis at one point, many points, or not at all if it's a horizontal line like we have in our current exercise.

Understanding where the line meets the y-axis is essential for graphing the equation and solving various algebraic problems. Identifying the y-intercept is straightforward: if you're given an equation, set \(x\) to zero and solve for \(y\). In our textbook problem, we found the y-intercept by using a known point and substituting it back into the equation, which confirmed that the y-intercept, \(b\), is -2. This tells us that every point on this line has a \(y\)-coordinate of -2, and we can select any value for \(x\) to find more points on this line.