Problem 75 - Precalculus Essentials

Question

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. $$(a, b) \text { and }(a, b+c)$$

Step-by-Step Solution

Answer

The slope of the line passing through the points \((a, b)\) and \((a, b+c)\) is undefined, indicating that the line is vertical.

Step 1: Identify Coordinates

The problem provides us with two points. The first point is \((a, b)\) and the second point is \((a, b+c)\). Here, the coordinates are \((x_1, y_1) = (a, b)\) and \((x_2, y_2) = (a, b+c)\).

Step 2: Calculate Slope

To calculate the slope m of the line passing through these points, we use the formula \(m = (y_2 - y_1) / (x_2 - x_1)\). In this scenario, since \(x_2 = x_1\), the denominator of the slope formula, \(x_2 - x_1\), is 0, resulting in a division by zero situation.

Step 3: Find Nature of Line

When the denominator of the slope is zero, it means the slope is undefined. This indicates that the line is vertical. As per convention, vertical lines do not have a defined slope.

Key Concepts

Vertical Line

When dealing with the concept of a vertical line, imagine a straight line that goes up and down, parallel to the y-axis on a graph. In our original exercise, we have two points: \((a, b)\) and \((a, b+c)\).
The x-coordinates of these points are the same, which means no matter how much the y-coordinates differ, the line doesn't slant left or right.
Key characteristics of vertical lines include:

• They are straight up and down and parallel to the y-axis.
• They do not cross the x-axis at any point, except possibly at infinity.
• Vertical lines are often described by equations of the form \(x = a\), where \(a\) is the constant value of \(x\).

In our given problem, since both x-coordinates of the points are \(a\), the line connecting them is indeed vertical.

Undefined Slope

The slope of a line is a measure that tells us how steep the line is. Slope is usually represented by \(m\) and calculated using the formula: \(m = \frac{(y_2-y_1)}{(x_2-x_1)}\).
However, in some cases, like in the exercise, you encounter a special scenario where the slope becomes undefined.
A slope becomes undefined when:

• The x-coordinates of both points are the same. This means the change in x, \(x_2-x_1\), equals zero.
• The denominator of the slope formula becomes zero, leading to an undefined mathematical condition.

In simple terms, an undefined slope indicates that the line does not have a "rise over run" measure because it's more of a rise without a run, directly pointing to a vertical line.

Division by Zero

Division by zero is a fundamental concept in mathematics that occurs when a number is divided by zero, such as in our slope calculation with \(m = \frac{(y_2-y_1)}{(x_2-x_1)}\). Where \(x_2-x_1\) equals zero, the division goes undefined.
Here's why division by zero is problematic:

• Mathematically, dividing by zero leads to a result that does not have a finite or calculable value, hence it remains undefined.
• It can be thought of as having zero width or being infinitely steep, reminding us of a vertical line's nature.

In practical terms, division by zero is avoided in calculations because it can cause errors in computations and diagrams, albeit reminding us again that the slope of a vertical line cannot be measured in conventional ways.