Problem 49 - Thinking Mathematically

Question

In Exercises 49-52, find the slope of the line passing through each pair of points or state that the slope is undefined. Assume that all variables represent positive real numbers. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. \((0, a)\) and \((b, 0)\)

Step-by-Step Solution

Answer

The slope of the line passing through the points \((0, a)\) and \((b, 0)\) is \(-a / b\), and the line falls from left to right.

Step 1: Identify the points

The question gives us two points, which are \((0, a)\) and \((b, 0)\). We can identify these two points as \((x1, y1)\) and \((x2, y2)\) respectively.

Step 2: Plug the values into the formula

Now, we can calculate the slope by plugging the corresponding values into the formula for slope: \(m = (y2 - y1) / (x2 - x1)\). By doing so, we get \(m = (0 - a) / (b - 0)\). Simplifying the numerator and the denominator gives us \(m = -a / b\).

Step 3: Determine the line's behaviour

If the slope is negative, the line falls from left to right. So our line falls since our slope is \(-a / b\), which is negative because both \(a\) and \(b\) are given as positive.

Key Concepts

Slope Formula

Understanding the slope of a line is crucial when studying linear equations and their graphs. The slope formula is a tool that helps us calculate the rate at which a line rises or falls. It is represented as \( m = \frac{y2 - y1}{x2 - x1} \), where \( (x1, y1) \) and \( (x2, y2) \) are two distinct points on the line.

To use the slope formula, subtract the y-coordinate of the first point from the y-coordinate of the second point, and divide this by the subtraction of the x-coordinate of the first point from the x-coordinate of the second point. This calculation can be seen as a ratio that expresses the vertical change per unit of horizontal change along the line.

For the exercise provided, plugging the coordinates \( (0, a) \) and \( (b, 0) \) into the formula helps us find the slope \( m = -\frac{a}{b} \) which is a simple yet fundamental application of the slope formula in understanding the geometry of linear graphs.

Undefined Slope

While most lines on a graph have a definable slope, there are special cases where a line might have what we call an undefined slope. This occurs when a line is vertical, which means it runs up and down the graph without any horizontal change. Since the slope formula involves dividing by the difference in x-coordinates, a vertical line's identical x-coordinates would lead to division by zero which is undefined mathematically.

If we were given points that had the same x-coordinate, say \( (c, d) \) and \( (c, e) \) where \( c \) is constant but \( d \) and \( e \) vary, our calculation for slope \( m = \frac{e - d}{c - c} \) results in a denominator of zero. Therefore, the slope is undefined. In the case of our exercise, the slope \( -\frac{a}{b} \) is defined because the x-coordinates of the two points are not the same, which allows for a valid calculation.

Slope Interpretation

Interpreting the slope of a line aids in understanding the relationship between variables and the general trend shown on a graph. A positive slope, obtained when the line rises from left to right, indicates an increasing relationship. A negative slope, as in \( -\frac{a}{b} \) from our exercise, means the line falls from left to right, representing a decreasing relationship between variables.

The magnitude of the slope determines the steepness of the line. A large absolute value indicates a steep line, whereas a small absolute value suggests a more gradual incline or decline. Additionally, a slope of zero corresponds to a horizontal line, depicting no change in the y-coordinate as the x-coordinate changes. This variety in line behaviour gives us insight into the way variables may impact one another within the context of the equation.