Problem 18 - Algebra 2

Question

Find the slope of the line that passes through each pair of points. $$ (4,-1.5),(4,4.5) $$

Step-by-Step Solution

Answer

The slope is undefined.

Step 1: Identify the Points

We start by identifying the coordinates of the two given points. The first point is \((x_1, y_1) = (4, -1.5)\) and the second point is \((x_2, y_2) = (4, 4.5)\).

Step 2: Use the Slope Formula

The slope \(m\) of the line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the given points, we have \[ m = \frac{4.5 - (-1.5)}{4 - 4} \]

Step 3: Calculate the Numerator

Calculate the change in the \(y\) values: \( 4.5 - (-1.5) = 4.5 + 1.5 = 6 \)

Step 4: Calculate the Denominator

Calculate the change in the \(x\) values: \( 4 - 4 = 0 \)

Step 5: Determine the Slope

Notice that the denominator is zero. This indicates that we are attempting to divide by zero, which is undefined in mathematics. This means the slope is undefined.

Key Concepts

Understanding Undefined Slope

When we talk about slope in mathematics, we are discussing how steep a line is. A slope measures how much a line rises or falls from left to right. Now, an undefined slope can be a tricky concept because it's basically when a line doesn't have a defined steepness.
This occurs when you try to divide by zero in the slope formula. Whenever the denominator of \( \frac{y_2 - y_1}{x_2 - x_1} \) becomes zero, the slope is undefined. This happens when we have two points with the same \( x \) coordinates. Hence, the line becomes undefined in terms of slope.

Working with Vertical Lines

A vertical line occurs when two points on a line share the same \( x \)-coordinate, as seen in our exercise. Vertical lines are a unique type of line because they go straight up and down rather than at an angle.
The slope of a vertical line is undefined because it requires dividing by zero in the slope calculation. Vertical lines can be easily identified on a graph as they do not "lean" towards either side, maintaining a strict vertical position. Whenever you spot that both \( x \) values are equal, you're dealing with a vertical line.

Intro to Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to study geometric figures using coordinates on a plane. It bridges algebra and geometry through graphs and equations of lines and shapes.
In our scenario, we used the coordinates \((4, -1.5)\) and \((4, 4.5)\) to analyze the relationship between the points. Coordinate geometry provides tools to calculate slopes, distances, and even angles using the points’ coordinates, making it an essential tool for understanding geometry at a deeper level.

Detailed Calculation Steps

To find the slope between two points, you need the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). It relies on the change in \( y \) ("rise") over the change in \( x \) ("run").
Here's the breakdown:

• Identify the coordinates of each point. In our example: \((x_1, y_1)\) is \((4, -1.5)\), and \((x_2, y_2)\) is \((4, 4.5)\).
• Substitute into the formula: calculate \( y_2 - y_1 \): \( 4.5 - (-1.5) = 6 \).
• Calculate \( x_2 - x_1 \): \( 4 - 4 = 0 \).
• The division by zero results in an undefined slope.

The steps clearly show how zero in the denominator leads to an undefined slope, making this line vertical.