Problem 31 - Algebra 1: Concepts and Skills

Question

ZERO OR UNDEFINED SLOPE Determine whether the slope is zero, undefined, or neither. $$ (6,2) \text { and }(9,2) $$

Step-by-Step Solution

Answer

The slope of the line that passes through the points (6,2) and (9,2) is zero.

Step 1: Calculate changes in x and y

Calculate change in x (Δx) and change in y (Δy) by subtracting the values of x and y coordinates of the respective points. The Δx is 9 - 6 = 3 and the Δy is 2 - 2 = 0.

Step 2: Find the slope

Use the formula for the slope, which is m = Δy/Δx. Therefore m = 0 / 3 = 0.

Step 3: Draw conclusion

Since the slope m=0, the slope is not undefined and the line is horizontal. So, the slope of the line passing through the points (6,2) and (9,2) is zero.

Key Concepts

Zero Slope

In mathematics, the slope of a line measures its steepness and direction. When we talk about zero slope, we refer to a very specific scenario. Imagine you're on a completely flat road; no uphill or downhill sections, just flat. That's what a zero slope represents—a line that doesn’t tilt up or down at all.

A zero slope occurs when the change in the y-values between two points on the line is zero. In other words, the y-coordinate doesn't change as you move along the line. This happens when you subtract the y-values of any two points on the line and get a result of zero.

• Formula: If \(m = \frac{\Delta y}{\Delta x}\) and \(\Delta y = 0\), then \(m = 0\).

Zero slope lines are always horizontal, which means they run parallel to the x-axis. They're easy to spot because they never rise or fall as you progress along them. This is the case with the points (6,2) and (9,2), where \(\Delta y = 0\), leading us to a zero slope.

Undefined Slope

An undefined slope is quite a different scenario from a zero slope. It happens when a line is perfectly vertical. Think of it as a steep cliff or a ladder standing straight up—no running horizontally at all.

The concept of undefined slope arises because the change in x-values (\(\Delta x\)) between two points on the line is zero. You can't divide by zero in the slope formula, \(m = \frac{\Delta y}{\Delta x}\), hence the term "undefined."

• Feature: A line with undefined slope runs parallel to the y-axis.

When trying to write an equation or interpret one with an undefined slope, you’ll find x takes on a constant value (e.g., x = 5). This means no matter where you look on the line, the x-value remains the same, unlike the line between (6,2) and (9,2) we explored earlier.

Horizontal Line

Horizontal lines are a cinch to understand! They are straight lines that run left to right across the graph without any vertical movement. This means they do not go up or down, staying at a constant y-value.

These lines are the perfect illustration of a zero slope. They appear as a flat surface on a graph, much like a calm lake or a quiet street without any hills.

• Equation: For any horizontal line, the equation is in the form y = b, where 'b' is a constant.
• Example: The line represented by points such as (6,2) and (9,2) is horizontal, giving us the equation y = 2.

Horizontal lines serve as great examples of how real-world flat surfaces appear on a graph. They're easy to work with because they don't change in height, making calculations straightforward and predictable.