Problem 73 - 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. \((0, a)\) and \((b, 0)\)

Step-by-Step Solution

Answer

The slope of the line passing through the points (0, a) and (b, 0) is -a/b. It falls as we move from left to right, and it is neither horizontal nor vertical.

Step 1: Calculate the Slope

The slope \( m \) between two points \( P1(x1, y1) \) and \( P2(x2, y2) \) can be found by using the slope formula \( m = (y2 - y1) / (x2 - x1) \). Here, our points are \( (0, a) \) and \( (b, 0) \). So, \( m = (0 - a) / (b - 0) = -a/b \)

Step 2: Determine the Line Orientation

A slope is defined as the rise over run, indicating how much the line rises or falls as we move from left to right. In this case, the slope is negative, so the line falls as we move left to right.

Step 3: Check for Horizontal or Vertical Line

It's important to note that horizontal lines have a slope of 0, and vertical lines have an undefined slope. Since our calculated slope is \( -a/b \), the line is neither horizontal nor vertical.

Key Concepts

Undefined Slope

When learning about the slope of a line, an essential concept to understand is the undefined slope. This occurs with vertical lines, where the x-coordinates for any two points on the line are the same, but the y-coordinates are different. Mathematically, if you attempt to calculate the slope using the slope formula, you’d have a denominator of zero, which is not permitted because division by zero is undefined in mathematics.

For example, consider a vertical line passing through the points (3, 5) and (3, -2). Using the slope formula, the slope (\( m \)) would be \( m = \frac{(-2) - (5)}{(3) - (3)} = \frac{-7}{0} \) which is undefined. This characteristic is fundamentally important, as it distinguishes vertical lines from all other lines that have slope values ranging from negative infinity to positive infinity.

Horizontal and Vertical Lines

Understanding the properties of horizontal and vertical lines is crucial. Horizontal lines run from left to right and have a constant y-coordinate. Their slope is 0 because there is no rise regardless of how much you 'run'; therefore, the line does not rise or fall. A classic example is the line passing through points (1, 4) and (5, 4), which would be horizontal.

In contrast, vertical lines go from top to bottom (or bottom to top) and have a constant x-coordinate. As mentioned earlier, the slope of these lines is undefined. It's important to distinguish between the two as they express different behaviors of a line graphically represented on a coordinate plane.

Slope Formula

The slope formula is a fundamental component for analyzing lines on a coordinate plane. It is defined as the ratio of the difference in the y-coordinates to the difference in the x-coordinates of two distinct points on a line. Expressed mathematically, the slope (\( m \)) between two points \( P1(x1, y1) \) and \( P2(x2, y2) \) is given by \( m = \frac{(y2 - y1)}{(x2 - x1)} \). This formula facilitates the determination of whether a line rises (positive slope), falls (negative slope), is flat (zero slope), or is vertical (undefined slope).

For instance, given the points (0, a) and (b, 0), applying the slope formula we get \( m = \frac{(0 - a)}{(b - 0)} = -\frac{a}{b} \), illustrating a slope that is not zero, positive, or undefined, which means the line would descend from left to right.

Positive Real Numbers

In the context of slopes and coordinate geometry, positive real numbers play a pivotal role. They consist of all the numbers greater than zero on the number line, without any imaginary or complex part. When discussing the slope of a line, positive slopes correspond to lines that rise from left to right. In practice, this means when you move from one point to another along the line, you move upwards while proceeding to the right.

Positive real numbers are often associated with increasing values in a linear relationship. Applying this to an exercise involving slope, if both variables represent positive real numbers, it suggests a context where you can expect growth or ascent, like profits over time or elevation over distance, provided the slope is positive.