Problem 32 - Algebra 1: Concepts and Skills

Question

ZERO OR UNDEFINED SLOPE Determine whether the slope is zero, undefined, or neither. $$ (5,-8) \text { and }(3,-8) $$

Step-by-Step Solution

Answer

The slope of the line that passes through the points (5,-8) and (3,-8) is zero.

Step 1: Identify the points

The given points are (5,-8) and (3,-8). Let's label them as follows: (x1,y1) = (5, -8) and (x2,y2) = (3, -8)

Step 2: Use the slope formula

The slope (m) of a line through two points (x1, y1) and (x2, y2) is calculated as m = (y2 - y1) / (x2 - x1). Plug in the values of the points into the formula: m = (-8 - (-8)) / (3 - 5).

Step 3: Simplify the expression

Simplify the values in the formula to get the slope: m = (0) / (-2) = 0

Step 4: Identify the type of slope

Since m = 0, the slope is zero. A slope of zero indicates a horizontal line.

Key Concepts

Understanding Undefined Slope

When learning about the characteristics of linear equations, the concept of an undefined slope is a critical one. An undefined slope occurs with a vertical line, where the x-coordinates of any two points on the line are the same, but the y-coordinates are different. This results in a division by zero in the slope formula, which is mathematically undefined. For example, if we have two points \( (x_1, y_1) \text{ and } (x_2, y_2) = (2, 3) \text{ and } (2, 7) \), applying the slope formula \(m = \frac{(y_2 - y_1)}{(x_2 - x_1)}\) would result in division by zero, hence an undefined slope. It's vital to recognize vertical lines as the graphical representation of undefined slopes; no matter how far up or down the line goes, it never moves horizontally.

Zero Slope Explained

The concept of zero slope is essential for understanding horizontal lines in Cartesian coordinates. When a line on a graph is horizontal, this means that there is no rise over the run—no change in the y-value no matter how much the x-value changes. This is represented by the slope formula \( m = \frac{(y_2 - y_1)}{(x_2 - x_1)} \) yielding a zero in the numerator, indicating that the change in y (the rise) is zero. For instance, if we look at the two points \( (5,-8) \text{ and } (3,-8) \), by plugging these into the slope formula the result is \( m = \frac{(-8 - (-8))}{(3 - 5)} = \frac{0}{-2} = 0 \). This confirms that the line has a zero slope and is, therefore, horizontal. It's a common mistake to assume that a flat line might have no slope, but in mathematics, it is precisely defined as a zero slope.

Slope Formula Demystified

To fully comprehend lines on a graph, one must understand the slope formula. The slope is a measure of how steep a line is and is calculated by the ratio of the 'rise' (the change in y) to the 'run' (the change in x) between any two points on the line. The formula for calculating the slope between two points \( (x_1, y_1) \text{ and } (x_2, y_2) \) is given by: \[ m = \frac{(y_2 - y_1)}{(x_2 - x_1)} \]

For a deeper understanding, let's walk through a practical application of the slope formula. Consider the points \( (5,-8) \text{ and } (3,-8) \) given in the original problem. To find the slope (m), we subtract the y-coordinate of the first point from the y-coordinate of the second point and divide the result by the difference between the x-coordinates of these points. In this case: \[ m = \frac{(-8 - (-8))}{(3 - 5)} = \frac{0}{-2} = 0 \]
The result tells us the line's slope is zero, indicating a horizontal line. Grasping the slope formula and how to apply it is not only crucial for solving equations but also for understanding the geometric interpretation of linear functions.