Problem 74 - College Algebra

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 either line through the points rises, falls, is horizontal, or is vertical. \((-a, 0)\) and \((0,-b)\)

Step-by-Step Solution

Answer

The slope of the line passing through the points is \( \frac{-b}{a} \) which means the line is falling.

Step 1: Identify the points

The provided points are \((-a, 0)\) and \((0, -b)\). These points can be identified as \(x_1 = -a\), \(y_1 = 0\), \(x_2 = 0\) and \(y_2 = -b\) respectively.

Step 2: Apply the slope formula

Using the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), substitute the values of the points to calculate the slope. Therefore, plug in the values: \(m = \frac{-b - 0}{0 -(-a)}\)

Step 3: Simplify the expression

Simplify the equation to obtain the slope: \(m = \frac{-b}{a}\)

Step 4: Determine the direction of the line

The slope \(m = \frac{-b}{a}\) is negative, means the line falls as we move from left to right.

Key Concepts

Calculating Slope

Calculating the slope of a line is essential in understanding how the line behaves within a two-dimensional space. To calculate the slope, we use the formula
\( m = \frac{y_2 - y_1}{x_2 - x_1}\).
This formula essentially measures the rate at which the line rises or falls as we move from one point to another horizontally. If we're given two points, let's say \((x_1, y_1)\) and \((x_2, y_2)\), we identify the vertical change by subtracting the y-coordinates, and the horizontal change by subtracting the x-coordinates.
For instance, in the exercise provided, the points were \((-a, 0)\) and \((0, -b)\), identifying as \(x_1 = -a\), \(y_1 = 0\), \(x_2 = 0\), and \(y_2 = -b\). When we substitute these into the slope formula, we get \(m = \frac{-b - 0}{0 - (-a)} = \frac{-b}{a}\). This calculation is crucial for understanding the line's direction, as will be discussed in the following sections.

Undefined Slope

An 'undefined slope' refers to a situation where a line is completely vertical, and hence doesn't have a defined slope. Mathematically, this occurs when the change in the x-coordinates is zero, because you end up with a division by zero in the slope formula, which is an undefined operation.
In the formula \( m = \frac{y_2 - y_1}{x_2 - x_1}\), if \(x_1 = x_2\), we get \( m = \frac{y_2 - y_1}{0}\), which is undefined.
A real-world example of undefined slope could be an elevator moving up or down in a building—it has vertical movement but no horizontal movement. Returning to our exercise example, the slope was \(m = \frac{-b}{a}\), implying that both a and b are nonzero and thus the slope is defined. If the problem instead presented points like (3, 2) and (3, -5), this would illustrate an undefined slope.

Direction of a Line

The direction of a line on a graph is indicated by whether it's rising, falling, horizontal, or vertical. The slope gives us this information:

• A positive slope means the line rises as it moves from left to right.
• A negative slope indicates the line falls as it moves from left to right.
• A zero slope means the line is horizontal.
• An undefined slope corresponds to a vertical line.

In our exercise, the slope was calculated as \(m = \frac{-b}{a}\), which tells us the slope is negative if a and b are positive. Thus, it means as one moves from left to right, the line falls. It is important to visualize this: starting from the left-highest point, the line will descend towards the right-lower direction. Knowing the direction helps in graphing linear equations and understanding the rate of change in physical phenomena like speed or current.