Problem 62 - Algebra A Combined Function

Question

Find the slope of the line containing each pair of points. (2,1),(2,-3)

Step-by-Step Solution

Answer

The slope is undefined as the line is vertical.

Step 1: Identify the Formula for Slope

To find the slope of a line given two points, we use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the given points.

Step 2: Assign the Points to the Variables

Assign the first point \((2, 1)\) to \((x_1, y_1)\) and the second point \((2, -3)\) to \((x_2, y_2)\). So, \( x_1 = 2 \), \( y_1 = 1 \), \( x_2 = 2 \), and \( y_2 = -3 \).

Step 3: Plug the Values into the Slope Formula

Substitute the values into the slope formula: \[ m = \frac{-3 - 1}{2 - 2} \].

Step 4: Simplify the Expression

Calculate the numerator and denominator: \(-3 - 1 = -4\) and \(2 - 2 = 0\). The expression becomes: \[ m = \frac{-4}{0} \].

Step 5: Interpret the Result

The division by zero in \( \frac{-4}{0} \) implies the slope is undefined, indicating a vertical line.

Key Concepts

Vertical Line

A vertical line is a unique type of line on a cartesian plane. Unlike horizontal or sloped lines, a vertical line runs straight up and down. All the points on a vertical line have the same x-coordinate, which means it doesn’t slant. This characteristic is why it doesn’t have a conventional slope like other lines.

When plotting a pair of points that form a vertical line, such as \[ (2,1) \text{ and } (2,-3) \],each point shares an x-coordinate of 2.

A vertical line through these points would be a straight line parallel to the y-axis. This can be visually represented by the line equation:

• \( x = 2 \)

This equation signifies that no matter the y-value, the x-value always remains constant at 2.

Undefined Slope

The slope of a line is generally a measure of how steep the line is, typically represented by the variable \( m \). For most lines, the slope can be defined using the rise over run principle: the change in y over the change in x.

However, when it comes to vertical lines, attempting to calculate the slope results in an undefined value. This occurs because a vertical line makes the denominator of the slope formula zero, as shown:

• Given the points \( (2,1) \) and \( (2,-3) \), the slope formula would look like \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 1}{2 - 2} = \frac{-4}{0} \).

As division by zero is mathematically undefined, the slope of a vertical line is therefore also undefined. This essential feature distinguishes vertical lines from other types of lines.

Finding Slope from Points

To find the slope of a line between two points, you start with the slope formula:

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

The process is relatively straightforward, where you subtract the y-value of the first point from the y-value of the second point, and do the same with the x-values.

However, if the x-coordinates of the points are the same, like in the case of the points \( (2,1) \) and \( (2,-3) \), the result is a zero denominator. This creates an undefined slope as the subtraction results in zero division, which is not defined in mathematics.

This exception underscores the importance of understanding vertical lines and their properties, where even though there is no conventional slope, our identification of it as undefined still provides valuable insight into the geometry of the line itself.