Problem 73 - Algebra for College Students

Question

What is the slope of a line that is perpendicular to a line with slope \(0.247 ?\)

Step-by-Step Solution

Answer

The slope of the perpendicular line is approximately \(-4.05\).

Step 1: Understand Slope of Perpendicular Lines

When two lines are perpendicular, the slope of one line is the negative reciprocal of the slope of the other line. This means if the slope of one line is given by \(m\), the slope of the line perpendicular to it is \(-\frac{1}{m}\).

Step 2: Apply the Negative Reciprocal Concept

The given slope is \(0.247\). To find the slope of the perpendicular line, take the negative reciprocal of \(0.247\). Calculate it as follows: \(-\frac{1}{0.247}\).

Step 3: Simplify the Negative Reciprocal

Now, simplify \(-\frac{1}{0.247}\) to find the slope of the perpendicular line. Perform the division: \(-\frac{1}{0.247} \approx -4.05\).

Key Concepts

slope calculation

To understand slope calculation, we must first recognize the slope as a measure of a line's steepness or incline. In math, slope is usually represented by the letter 'm'. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. Mathematically, this is expressed as \(m = \frac{\Delta y}{\Delta x}\), where \(\Delta y\) is the change in the y-coordinates and \(\Delta x\) is the change in the x-coordinates.
For example, given two points (1,2) and (3,4), the slope \ would be calculated as:
\[ m = \frac{4-2}{3-1} = \frac{2}{2} = 1 \] This means that for every unit increase in the x-coordinate, the y-coordinate increases by 1. Understanding this concept is critical for solving problems involving perpendicular lines.

negative reciprocal

In the context of perpendicular lines, one of the most important things to understand is the concept of the negative reciprocal. When two lines are perpendicular, their slopes are related in a special way: the slope of one line is the negative reciprocal of the slope of the other line.
To find the negative reciprocal of a given slope, you first take the reciprocal (flip the fraction) and then change the sign. For instance, if the slope of a line is represented by \(m\), the slope of a line perpendicular to it will be \ \( -\frac{1}{m} \).
Let's apply this. Given that the slope of a line is 0.247, the slope of the line perpendicular to it is calculated as:
\[ -\frac{1}{0.247} \] Perform the division to simplify:
\[ -\frac{1}{0.247} \approx -4.05 \]
So, the slope of the line perpendicular to the given slope is approximately -4.05.

linear equations

Linear equations represent straight lines on a graph and can be written in various forms, such as slope-intercept form, which is \(y = mx + b\), where 'm' is the slope and 'b' is the y-intercept.
The slope-intercept form is particularly useful because it allows you to easily identify the slope and y-intercept of the line. For example, if you have the equation \( y = 2x + 3 \), you can see that the slope 'm' is 2, and the y-intercept 'b' is 3.
Finding the equation of a line perpendicular to a given line involves using the negative reciprocal of its slope. If the original line has a slope of 0.247 and passes through a point (let's say (1,2)), the perpendicular line would have a slope of -4.05. Using the point-slope form \( y - y_1 = m(x - x_1) \) and substituting in, we'll get:
\[ y - 2 = -4.05(x - 1) \]
This allows us to describe the linear relationship between the x and y coordinates along the perpendicular line.