Problem 51 - Thinking Mathematically

Question

Find the slope of the line passing through each pair of points or state that the slope is undefined. Assume that all variables represent positive real numbers. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. \((a, b)\) and \((a, b+c)\)

Step-by-Step Solution

Answer

The slope of the line passing through the points (a, b) and (a, b+c) is undefined. The line is vertical as dividing by zero is undefined.

Step 1: Identify the coordinates of the points

Here the two points are \( (a, b) \) and \( (a, b+c) \). So we can identify the coordinates as follows: \( x_1 = a, y_1 = b, x_2 = a, y_2 = b+c \)

Step 2: Use the slope formula

To calculate the slope of the line passing through the two points, we use the formula: \( slope = (y_2 - y_1) / (x_2 - x_1) \). Here, applying the values, we get: \( slope = (b+c - b) / (a - a) \)

Step 3: Simplify the equation

Solving the equation gives us \( slope = c / 0 \)

Step 4: Understanding the result

Dividing anything by 0 is undefined, thus the slope of the line is undefined. That means the line is vertical.

Key Concepts

Slope of the Line

The slope of a line in coordinate geometry is a measure of how steep the line is. It is calculated as the ratio of the change in the y-coordinate to the change in the x-coordinate between two points on a line. In simpler terms, it tells us how much the line goes up or down as we move from left to right across a graph.

The formula to calculate the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]
When you have two points, to find the slope, you essentially subtract the y-coordinate of the first point from the y-coordinate of the second point and divide the result by the subtraction of the x-coordinate of the first point from the x-coordinate of the second point. A positive slope indicates that the line rises from left to right, while a negative slope indicates that the line falls from left to right. A slope of zero means the line is horizontal.

Undefined Slope

An undefined slope occurs when the line is perfectly vertical. In mathematical terms, this happens when the x-coordinates of two points on the line are the same, which creates a division by zero in the slope formula.

Take the points from the original exercise, \( (a, b) \) and \( (a, b+c) \) - since \( x_1 = x_2 = a \) the formula for the slope would appear as:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{b+c - b}{a - a} = \frac{c}{0} \]
Division by zero is impossible in standard arithmetic, which is why we say the slope is undefined. This signals that the line does not have a consistent rate of change since it goes straight up and down.

Vertical Line Graphing

When graphing a vertical line, one must understand that this type of line crosses the y-axis at a point but never crosses the x-axis. All points on a vertical line have the same x-coordinate but different y-coordinates.

In the exercise provided, since both points \( (a, b) \) and \( (a, b+c) \) share the same x-coordinate, the graph will be a straight line that intersects the x-axis at \( x = a \) and runs parallel to the y-axis. This is represented by the equation \( x = a \) for the line.

Because a vertical line has an undefined slope, it neither rises nor falls, and the concept of slope does not apply in the conventional sense. The inability to determine the slope underscores the unique nature of vertical lines in comparison to non-vertical lines, which can have slopes that are positive, negative, or zero.