Problem 44 - College Algebra with Corequisite Support

Question

For the following exercises, find the slope of the line that passes through the given points. (3,-2) and (3,-2)

Step-by-Step Solution

Answer

The slope is undefined because we cannot divide by zero.

Step 1: Identify the Points

We are given two points: \((3, -2)\) and \((3, -2)\). Here, both points are exactly the same.

Step 2: Determine the Slope Formula

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

Step 3: Substitute the Points into the Formula

Substitute the coordinates of the points \((3, -2)\) and \((3, -2)\) into the slope formula. This gives us:\[ m = \frac{-2 - (-2)}{3 - 3} \]

Step 4: Calculate the Slope

Simplify the expression:\[ m = \frac{-2 + 2}{0} = \frac{0}{0} \]This results in an undefined or indeterminate form, since division by zero is not possible.

Key Concepts

Points Identification

When you need to find the slope of a line, the first essential step is to identify the coordinates, or points, through which the line passes. Let's say you are given two points, like

• \((3, -2)\)
• \((3, -2)\)

From this, each point consists of an \(x\)-coordinate and a \(y\)-coordinate. In this case, both points are the same. This is a unique situation in which the line doesn't extend in any direction because both points overlap completely.

Slope Formula

Once you've identified your points, the next step is to use the slope formula. The slope of a line, often represented by the letter \(m\), describes its steepness and direction. The formula to find the slope of a line passing through two points, \((x_1, y_1)\) and \((x_2, y_2)\), is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]This formula calculates the change in \(y\) divided by the change in \(x\), essentially finding out how long the vertical distance is in relation to the horizontal distance between the two points.

Undefined Slope

Sometimes, calculating the slope can result in an undefined expression. This occurs when the change in \(x\) (the denominator in the slope formula) is 0. For example, when substituting the points \((3, -2)\) and \((3, -2)\) into the formula:\[m = \frac{-2 - (-2)}{3 - 3}\]The expression \[m = \frac{0}{0}\]is reached. Since this form doesn’t give a value, it's called an undefined slope, meaning the line is vertical or doesn't have a slope as traditionally calculated.

Division by Zero

In mathematics, division by zero is not defined. This means if you have zero as the denominator in a fraction, the expression does not yield a meaningful number or result. In our example:\[m = \frac{0}{0}\]Here, since both the numerator and denominator are zero, this creates an indeterminate form. This situation arises in scenarios where the two points are identical, as in this case. The line does not actually "rise" or "run" anywhere, leading to a situation where slope cannot be determined in the usual way.