Problem 10 - A Graphical Approach to Precalculus with Limits

Question

Why is it not possible to write a slope-intercept form of the equation of the line through the points \((12,6)\) and \((12,-2) ?\)

Step-by-Step Solution

Answer

The line is vertical, so it lacks a defined slope for slope-intercept form.

Step 1: Identify the Points

The given points are \((12, 6)\) and \((12, -2)\). We need to find the slope between them.

Step 2: Calculate the Slope

The formula for slope is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Plug the coordinates into the formula: \( m = \frac{-2 - 6}{12 - 12} = \frac{-8}{0} \).

Step 3: Interpret the Undefined Slope

Since division by zero is undefined, a slope of \(\frac{-8}{0}\) means the slope is undefined. This occurs when a line is vertical.

Step 4: Explain the Slope-Intercept Form

The slope-intercept form of a line is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Vertical lines cannot have this form because they don't have a defined slope.

Step 5: Conclude the Explanation

Thus, it is not possible to express the equation of a line through these points in slope-intercept form since the line is vertical. Vertical lines are expressed in the form \( x = k \), where \( k \) is the constant x-coordinate for any point on the line.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a useful way to describe a line with a simple equation. This form is written as \( y = mx + b \), where:

• \( m \) represents the slope of the line
• \( b \) is the y-intercept, the point where the line crosses the y-axis

This equation is particularly handy because it allows you to quickly see the slope and y-intercept of the line. However, one assumption exists: the line must have a defined slope. In cases where the line is vertical, like in the given exercise, the slope is undefined. This means we can't use \( y = mx + b \) to describe vertical lines.

Undefined Slope

An undefined slope occurs when looking at vertical lines. To calculate the slope, use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). If you find yourself dividing by zero, the slope is undefined. For example, in the exercise, a vertical line is between the points \((12, 6)\) and \((12, -2)\). Both x-coordinates are the same, leading to division by zero: \( m = \frac{-8}{0} \). Since the slope is undefined, it indicates a vertical line. Such vertical lines rise straight up and down, unable to slant or have a slope, making it impossible to fit them into a standard slope-intercept form.

Equation of a Line

An equation of a line tells us the relationship between the x and y coordinates along that line. The most common form is the slope-intercept form \( y = mx + b \), readily showing slope and y-intercept. However, vertical lines are best expressed differently. When a line is vertical, every point on it has the same x-coordinate. Therefore, the equation of such a line is in the form \( x = k \), where \( k \) is the constant x value. In the exercise, the equation for the line through points \((12, 6)\) and \((12, -2)\) is simply \( x = 12 \). This equation reflects the unchanging nature of the x-value along the vertical line.

Division by Zero

Division by zero is a mathematical operation that can't be completed, as it is undefined. Consider it as trying to divide a given number among no groups, resulting in an undefined state. In slope calculations, particularly in the basic formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), there can be an attempt to divide by zero when the x-values are the same. This leads to an undefined slope, signaling a vertical line. Beware of division by zero, as it indicates scenarios that traditional methods can't handle, such as the determination of slope for vertical lines. It's a significant concept to remember when analyzing and plotting lines.