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Chapter 11: Problem 275

Why is the slope of a vertical line undefined?

Short Answer

Expert verified
A vertical line has an undefined slope because division by zero in the slope formula results in an undefined value.

Step by step solution

01

- Identify the Structure of a Vertical Line

A vertical line has the same x-coordinate for all points. For example, a vertical line passing through x = 3 can be represented by the equation x = 3. This means that every point on this line has an x-coordinate of 3, regardless of the y-coordinate.
02

- Apply the Slope Formula to a Vertical Line

When selecting two points on a vertical line, say (3, y_1) and (3, y_2), we observe that the x-coordinates are the same (x_1 = x_2 = 3). Substituting these points into the slope formula, we get: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{y_2 - y_1}{3 - 3} = \frac{y_2 - y_1}{0} \]. This results in division by zero, which is undefined in mathematics.
03

- Conclusion

Since the denominator in the slope formula for a vertical line is zero, the slope of a vertical line is undefined because division by zero is not possible.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertical Line
A vertical line is a unique type of line in geometry. Unlike other lines that can be slanted at various angles, a vertical line runs straight up and down.
What sets a vertical line apart is that it maintains a constant x-coordinate for all its points. For instance, if we consider a vertical line represented by the equation x = 3, then every point on this line has an x-coordinate of 3, regardless of its y-coordinate.
This means any point on the line will look like (3, y), where 'y' can be any value.
Slope Formula
The slope of a line is a measure of its steepness and is calculated using the slope formula.
The slope formula is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Where:\
  • \( m \) is the slope of the line
  • \( (x_1, y_1) \) and \( (x_2, y_2) \) are coordinates of two points on the line
By plugging the values of the coordinates into this formula, we can determine the slope, which tells us how much the 'y' value changes for a unit change in 'x'.
Division by Zero
Division by zero is a key mathematical concept that is not allowed.
In the context of the slope formula, if the x-coordinates of two points are the same (as in the case of a vertical line), the denominator becomes zero.
Let's elaborate: For a vertical line represented by the points \((3, y_1)\) and \(( 3, y_2)\), the slope formula becomes
\[ m = \frac{y_2 - y_1}{3 - 3} = \frac{y_2 - y_1}{0} \]
This results in a division by zero, which is undefined in mathematics because we cannot divide any number by zero.
x-coordinate
The x-coordinate is the first part of an ordered pair (x, y) that represents a point on the coordinate plane.
It tells us how far to move horizontally from the origin (0, 0).
In a vertical line, this x-coordinate is the same for all points.
For instance, in the vertical line equation x = 3, the x-coordinate will always be 3.
This uniform x-coordinate is the reason why the denominator in the slope formula for a vertical line is zero, resulting in an undefined slope.
y-coordinate
The y-coordinate is the second part of an ordered pair (x, y) that specifies a point's position on the coordinate plane.
It indicates how far to move vertically from the origin (0, 0).
In the context of a vertical line, the y-coordinate can vary while the x-coordinate remains constant.
This variability ensures that even though the x-coordinate is fixed, multiple points can exist on a vertical line due to different y-coordinates.

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