Problem 13 - Applied Calculus

Question

Find the slope (if it is defined) of the line determined by each pair of points. $$ (2,-1) \text { and }(2,5) $$

Step-by-Step Solution

Answer

The slope is undefined (the line is vertical).

Step 1: Understanding the problem

To find the slope of a line through two points, we use the formula for the slope: \( m = \frac{y_2-y_1}{x_2-x_1} \). Given the points \((2, -1)\) and \((2, 5)\), our task is to substitute these coordinates into this formula to find the slope.

Step 2: Substitute the coordinates into the slope formula

We assign the points: \((x_1, y_1) = (2, -1)\) and \((x_2, y_2) = (2, 5)\). Substitute these into the slope formula: \( m = \frac{5 - (-1)}{2 - 2} \).

Step 3: Calculate the difference in y-values

The difference in the y-values is \(5 - (-1) = 5 + 1 = 6\). So, the formula becomes \( m = \frac{6}{2 - 2} \).

Step 4: Calculate the difference in x-values

The difference in the x-values is \(2 - 2 = 0\). This makes the slope formula \( m = \frac{6}{0} \).

Step 5: Interpret the result

Since division by zero is undefined in mathematics, the slope of a vertical line is also undefined. Therefore, the slope of the line passing through the points \((2, -1)\) and \((2, 5)\) is undefined.

Key Concepts

Slope of a Line

The slope of a line is a measure that describes the direction and steepness of the line. It is a fundamental concept in calculus and geometry. The slope is calculated using two points on the line:

• Point 1: - Coordinates are \( (x_1, y_1) \)
• Point 2: - Coordinates are \( (x_2, y_2) \)

The formula to determine the slope \( m \) is given by:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This formula calculates the ratio of the change in the vertical direction \( (y_2 - y_1) \) to the change in the horizontal direction \( (x_2 - x_1) \). If the slope \( m \) is positive, the line rises from left to right.If \( m \) is negative, the line falls from left to right. If the slope \( m \) is zero, the line is horizontal.Let's look at the case when the slope becomes undefined.

Undefined Slope

An undefined slope occurs when the difference in the x-values of the two points is zero. This happens because we would divide by zero when using the slope formula. Recall the slope formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \] An undefined slope can be understood as:

• Whenever \( x_1 = x_2 \), the denominator \( (x_2 - x_1) \) equals zero.
• Division by zero in mathematics is undefined.

Thus, the slope of a line becomes undefined. In practical terms, it means if you try to measure how steep the line is, it's not possible because there's no horizontal change, only vertical. This leads us to our next concept, the vertical line.

Vertical Line

A vertical line in the coordinate plane is a line where all points have the same x-coordinate. This characteristic leads to an undefined slope because there’s no horizontal movement between any two points on the line. For a vertical line:

• All points share the same x-value.
• There is no change in x-values (zero horizontal change).

Therefore, when you attempt to calculate the slope with points \( (x_1, y_1) = (2, -1) \) and \( (x_2, y_2) = (2, 5) \), the x-values are identical. This results in an undefined slope, representing a perfectly vertical line. Moreover, vertical lines are unique because they do not have a "steepness" to measure in the traditional sense, which is why vertical lines are often a common topic of discussion in understanding slopes.