Problem 31 - Beginning and Intermediate Algebra

Question

Use the slope formula to find the slope of the line containing each pair of points. $$(9,0) \text { and }(9,4)$$

Step-by-Step Solution

Answer

The line containing the points \((9,0)\) and \((9,4)\) is a vertical line, and its slope is not defined as it would result in a division by 0.

Step 1: Identify the coordinates of the points

Our given pair of points are \((9,0)\) and \((9,4)\). We will label them as follows: - Point 1: \((x_1, y_1) = (9,0)\) - Point 2: \((x_2, y_2) = (9,4)\)

Step 2: Apply the slope formula

Now, we will apply the slope formula: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ Substituting the coordinates of the points into the formula, we get: $$m = \frac{4 - 0}{9 - 9}$$

Step 3: Simplify the expression

Now, we will simplify the expression to find the slope: $$m = \frac{4}{0}$$ However, a fraction with 0 in the denominator is not defined. Therefore, we cannot find the slope in this case, which means the line containing these points must be vertical.

Step 4: Conclusion

Since the slope is not defined due to a division by 0, the line containing the points \((9,0)\) and \((9,4)\) is a vertical line.

Key Concepts

Vertical Line

A vertical line is a line that runs straight up and down. It has a distinctive characteristic where every point on the line has the same x-coordinate.
For example, the points

• (9, 0)
• (9, 4)

are on a vertical line because both points share the x-coordinate of 9.
This vertical alignment means that the line is parallel to the y-axis.
Vertical lines are unique compared to other lines as they cannot be represented by the usual slope-intercept form of a line equation, which makes them special when we discuss slopes.

Undefined Slope

The concept of slope tells us how steep a line is. For most lines, you can calculate this using the rise over run formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]
However, vertical lines don't have a defined slope. Why? Because the formula requires you to divide by

• the difference in the x-coordinates (\(x_2 - x_1\))

and for vertical lines, this difference is zero.
Mathematically, division by zero is undefined, so vertical lines have what's called an undefined slope.
This means we cannot quantify their steepness like we do with other lines.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, relates algebra and geometry through graphs and coordinates.
It uses the coordinate plane where each point is defined by a pair of numbers

• (x, y)

These coordinates help in plotting points, lines, and shapes for visualization and analysis.
In our exercise, the points

• (9, 0)
• (9, 4)

lie on a vertical line in coordinate geometry because they share the same x-coordinate.
This powerful tool allows us to solve problems using both numeric calculations and spatial reasoning.

Slope Formula

The slope formula helps to determine the steepness and direction of a line.
Given two points, \((x_1, y_1)\) and \((x_2, y_2)\), it's expressed as: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]
This formula uses the change in y (vertical change) over the change in x (horizontal change).
In this exercise, substituting the points

• (9, 0)
• (9, 4)

into the formula results in a division by zero, leading to an undefined slope.
The slope formula is crucial in various applications, helping to describe linear relationships and predict trends.