Problem 43 - A Graphical Approach to College Algebra

Question

Find the slope (if defined) of the line that passes through the given points. $$(-11,3) \text { and }(-11,5) \quad $$

Step-by-Step Solution

Answer

The slope is undefined; the line is vertical.

Step 1: Recall the formula for the slope of a line

The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

Step 2: Substitute the given points into the formula

Using the points \((-11, 3)\) as \((x_1, y_1)\) and \((-11, 5)\) as \((x_2, y_2)\), substitute them into the formula:\[m = \frac{5 - 3}{-11 - (-11)} = \frac{2}{-11 + 11} = \frac{2}{0}\]

Step 3: Analyze the slope calculation

The denominator in the slope calculation is \(0\), which means the slope is undefined because division by zero is not possible in mathematics. This indicates the line is vertical.

Key Concepts

Undefined Slope

In mathematics, an undefined slope occurs when a line is vertical. Imagine standing on the ground and looking straight up; that's similar to a vertical line in geometry. When you calculate the slope of a line with two given points, you may sometimes end up dividing by zero. This happens specifically when both points share the same x-coordinate. Remember, division by zero is not defined in math—it's like trying to split something into zero equal parts. Thus, the slope is undefined when the line is vertical. Vertical lines have a distinct characteristic: they go up and down but not side to side.

Vertical Line

A vertical line is a special type of line on the Cartesian plane. It rises straight up and down, parallel to the y-axis, and does not tilt in any other direction. In the coordinate plane:

• Every point on the line shares the same x-coordinate.
• The line's equation is in the form of \(x = a\), where \(a\) is the constant x-coordinate of every point on the line.

This means that for the points (-11, 3) and (-11, 5), the line is vertical and is represented by the equation \(x = -11\). Vertical lines do not have a slope that can be expressed numerically. Instead, their slope is undefined. Understanding the nature of vertical lines is crucial in visualizing and interpreting line graphs.

Slope Formula

The slope formula is a fundamental tool used to determine the steepness or incline of a line. When given two points on a line, \((x_1, y_1)\) and \((x_2, y_2)\), the slope \(m\) is calculated using the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]This formula measures how much "rise" there is for a given "run" between two points. However, if the denominator is zero, as seen when \(x_1\) equals \(x_2\), the slope becomes undefined. This is relevant when dealing with the exercise points (-11, 3) and (-11, 5), leading to an undefined slope because there is no horizontal change. Understanding the slope formula helps in analyzing the properties and orientation of different lines.