Problem 75 - Elementary Algebra

Question

If one line has a slope of \(\frac{2}{3}\), and another line has a slope of 2 , which line is steeper? Explain your answer.

Step-by-Step Solution

Answer

The line with a slope of 2 is steeper.

Step 1: Understand the Concept of Slope

The slope of a line is a measure of its steepness. A larger slope means a steeper line, while a smaller slope means a less steep line. The slope is calculated as 'rise over run', i.e., the change in the y-coordinate divided by the change in the x-coordinate.

Step 2: Compare the Slopes

To determine which line is steeper, compare the slopes given: \( \frac{2}{3} \) and 2. The slope of \( \frac{2}{3} \) corresponds to a rise of 2 for every run of 3, while the slope of 2 corresponds to a rise of 2 for every run of 1.

Step 3: Analyze the Values

\( \frac{2}{3} \) is equivalent to approximately 0.67 when written in decimal form, and the slope of 2 remains as 2. Comparing these numbers, 2 is greater than 0.67.

Step 4: Determine Which Line is Steeper

Since 2 is greater than 0.67, the line with a slope of 2 is steeper than the line with a slope of \( \frac{2}{3} \).

Key Concepts

Line Comparison

When you compare two lines, the main factor to consider is the slope, which tells us about their steepness. Here, we are working with two lines, each defined by a different slope. The problem at hand asks us to decide which line is steeper: one line with a slope of \( \frac{2}{3} \) and another with a slope of 2.
In line comparison, we observe the numerical value of the slope. A greater numerical value indicates a steeper line. For example:

• Slope \( \frac{2}{3} \) translates into 0.67 in decimal form.
• The slope 2 remains as 2 in decimal form.

From this, it's clear: the higher the number, the steeper the slope. Therefore, without needing to graph these lines, we can confidently conclude that the line with a slope of 2 is steeper than the one with a slope of 0.67.

Steepness

When talking about the steepness of a line, we refer to the slope, which is the incline or decline of a line on the coordinate plane. The steeper the line, the larger the slope value. Steepness helps us understand how quickly a line ascends or descends as it moves from left to right across the graph.
The slope of a line, in essence, reveals its steepness through numerical values:

• Steeper lines have larger positive slope values, meaning they rise more sharply.
• Lines that are less steep have smaller slope values, showing a more gentle rise.

In our given problem, a slope of 2 indicates a more dramatic ascent as opposed to \( \frac{2}{3} \), signifying that the line with slope 2 stands at a steeper angle compared to the line with slope \( \frac{2}{3} \). This concept is integral for analyzing the configuration and steepness of lines on a plane.

Rise Over Run

The concept of "rise over run" is essential to understanding what slope means. "Rise" refers to the vertical change between two points on a line, while "run" is the horizontal change between those points.
The formula to calculate slope is given by:\[\text{slope} = \frac{\text{rise}}{\text{run}}\]In our exercise, when examining slopes:

• For the slope \( \frac{2}{3} \), the rise is 2 units, and the run is 3 units, meaning for every 3 units you move horizontally, the line rises 2 units.
• For the slope of 2, the rise is 2, and the run is 1, resulting in a line that ascends 2 units for every 1 unit of horizontal movement.

Simply put, the smaller the run compared to the rise, the steeper the line. That's why a rise-over-run ratio of 2 (with a run of 1) shows a steeper incline than a ratio of \( \frac{2}{3} \) (with a run of 3). This straightforward explanation of rise over run provides a clear picture of how slope dictates the angle of a line.