Problem 7 - Algebra A Combined Function

Question

Find the slope of the line that passes through the given points. See Examples 1 and 2. $$ (-4,3) \text { and }(-4,5) $$

Step-by-Step Solution

Answer

The slope of the line is undefined.

Step 1: Recall the formula for slope

The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. This formula gives us the rate of change of the line.

Step 2: Substitute the given points into the formula

Substitute the points \((-4, 3)\) and \((-4, 5)\) into the slope formula. Here, \(x_1 = -4\), \(y_1 = 3\), \(x_2 = -4\), and \(y_2 = 5\). Therefore, the slope \( m \) is calculated as: \[ m = \frac{5 - 3}{-4 - (-4)} = \frac{2}{0} \].

Step 3: Analyze the result

The calculation leads to a division by zero: \(\frac{2}{0}\). This means the slope is undefined, which indicates that the line is vertical.

Key Concepts

Slope Formula

The slope formula is an essential tool in algebra and geometry used to determine the steepness or the direction of a line. It describes how much a line rises or falls as it moves from left to right. The formula is expressed as:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

In this formula, \( m \) represents the slope, and the coordinates \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on the line.
This formula calculates the change in \( y \) (the vertical difference) divided by the change in \( x \) (the horizontal difference).
A positive slope indicates an uphill line from left to right, while a negative slope indicates a downhill direction.
A zero slope means the line is perfectly horizontal.

Vertical Line

A vertical line in geometry is a line that goes straight up and down. It doesn't tilt in any direction. When you come across a line where all the \( x \)-coordinates are the same for each point, you have a vertical line. For example, points like \((-4, 3)\) and \((-4, 5)\) lie on a vertical line because the \( x \)-value is \(-4\) for both points.
Vertical lines are unique because they do not have a well-defined slope using the standard formula. Instead, the slope is undefined due to the nature of dividing by zero, which we'll touch on next.
Think about a vertical line as being like a standing wall; it doesn't slant, it simply rises from the ground.

Undefined Slope

An undefined slope occurs in the context of the slope formula when you end up needing to divide by zero.
This happens specifically when the \( x \)-coordinates of both points on the line are the same. For our example with points \((-4, 3)\) and \((-4, 5)\), substituting these into the formula gives:

• \( m = \frac{5 - 3}{-4 - (-4)} = \frac{2}{0} \)

Because of the division by zero, we cannot calculate a finite number. Hence, the slope is termed "undefined."
This is common for vertical lines. When you encounter an undefined slope, remember it's an indicator that the line in question is vertical, highlighting its distinct characteristic from horizontal, slanted, or other lines with finite slopes.