Problem 67 - Algebra 1

Question

MULTIPLE CHOICE The slope of the line passing through the points \((0,5)\) and \((0,-3)\) is \(\underline{?}\) \(\begin{array}{llll}\text { (A) positive } & \text { (B) negative } & \text { (C) zero }\end{array}\) (D) undefined

Step-by-Step Solution

Answer

The slope of the line passing through given points is undefined, so the answer is (D)

Step 1: Identify the given points

The points through which the line passes are \( (0,5) \) and \( (0,-3) \). Let's denote the first point by \( (x1,y1) = (0,5) \) and the second point by \( (x2,y2) = (0,-3) \).

Step 2: Substitute into the slope formula

The formula for slope of a line passing through two points \( (x1,y1) \) and \( (x2,y2) \) is \( m = \frac{y2 - y1}{x2 - x1} \). Substituting our values, we obtain \( m = \frac{-3 - 5}{0 - 0} \).

Step 3: Simplify the equation

Simplifying the above equation leads to a division by zero situation \( m = \frac{-8}{0} \). Division by zero is undefined in mathematics.

Key Concepts

Undefined Slope

When we talk about the slope of a line, we often describe how steep the line is on a graph.
However, sometimes the slope can be undefined. This happens in particular situations. In geometry, an undefined slope occurs when we try to describe a vertical line. Imagine no horizontal movement between two points—just a straight up or down direction.
In math terms, this means that the change in x-values is zero. When we meet these vertical lines, attempting to calculate the slope becomes problematic. Slope is determined using the ratio of the change in y-values to the change in x-values.
For vertical lines, because our change in x-values is zero, the slope becomes undefined. This is a fundamental rule in mathematical definitions.

Slope Formula

The slope formula is the tool we use to discover the slope of a line. Let's break it down for clarity.The slope formula is written as:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Here,

• \((x_1, y_1)\) and \((x_2, y_2)\) represent two points on the line.
• \(y_2 - y_1\) measures the vertical change, or how much the line goes up or down.
• \(x_2 - x_1\) measures the horizontal change, or how much the line goes left or right.

By dividing the vertical change by the horizontal change, we determine the slope.
It gives us a number that indicates whether the line inclines upwards, declines downwards, or remains flat.

Division by Zero

The concept of division by zero can be puzzling. It's important to understand why it's such a unique case in math.
Simply put, division by zero is mathematically undefined. This means there's no sensible answer.Here's why: suppose you have any number divided by zero, like \( \frac{8}{0} \).
You might attempt to ask how many times zero can "fit" into eight. But because zero signifies nothingness, you can't find a real number that answers this.
Even trying to calculate ends up contradicting mathematical consistency.In practice, any division by zero, such as in the slope calculation for a vertical line, results in undefined values.
Therefore, we use terms and conventions, like "undefined slope," to communicate these scenarios in math effectively.