Problem 10 - A Graphical Approach to College Algebra

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 slope is undefined for a vertical line, which doesn't fit in slope-intercept form.

Step 1: Identify the slope formula

The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). We will use this formula to find the slope between the points \((12, 6)\) and \((12, -2)\).

Step 2: Plug in the values

Substitute the values of the points into the slope formula: \( m = \frac{-2 - 6}{12 - 12} = \frac{-8}{0} \).

Step 3: Evaluate the slope

The slope \( m = \frac{-8}{0} \) is undefined because division by zero is not defined. This indicates a vertical line.

Step 4: Review slope-intercept form

The slope-intercept form of a line is \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept. For a vertical line, the slope is undefined, meaning it cannot be expressed in slope-intercept form.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a simple way to write the equation of a straight line. It is written as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept—the point where the line crosses the y-axis. This form is incredibly useful because it allows you to quickly identify the slope and y-intercept, helping to easily draw the graph of the line.
For most lines, the slope-intercept form is perfect. However, there is an exception: vertical lines. Why? Vertical lines have an undefined slope, making it impossible to plug a value for \( m \) into the equation. Without a defined slope, the slope-intercept equation doesn't work for vertical lines.
In cases where you encounter a vertical line, it is better to use a different form, such as \( x = a \) where \( a \) is the x-coordinate of any point through which the line passes. This keeps everything simple and avoids confusion.

Slope Formula

The slope formula is crucial when you want to find the slope of a line given two points. The formula is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). It tells us how much the line rises or falls as it moves from one point to the next. Essentially, the slope describes the line's steepness.
Let's see how it works by applying it to two points we have. Using the points \((12, 6)\) and \((12, -2)\), you plug these values into the formula: \( m = \frac{-2 - 6}{12 - 12} = \frac{-8}{0} \). See something strange? The denominator is zero. In mathematics, you can't divide by zero—it means the slope is undefined.
This undefined slope tells us something special: the line is vertical. When both \( x \) values are identical for different points but the \( y \) values differ, the result is a vertical line. This is one of those special cases where using the slope formula tells us more than just how steep the line is—it shows us the line's orientation.

Undefined Slope

An undefined slope is a unique concept that occurs when dealing with vertical lines. In a typical linear equation, the slope is defined by how much the line goes up or down for each unit it moves horizontally. This concept is perfectly fine when there is horizontal movement.
But consider points where there is no horizontal movement, like \((12, 6)\) and \((12, -2)\). Here, the x-values are the same (12), meaning the line doesn't move left or right at all—it's a straight-up-and-down line, or vertical line.
Such lines have an undefined slope because calculating the slope involves a division by zero in the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). The division by zero makes the slope undefined, a mathematical condition indicating the impossibility of representing the slope in a numerical form.
To express this line, we instead use \( x = 12 \), highlighting its vertical nature without getting into complications of dividing by zero.