Get started for free
Log In Start studying!
Get started for free Log out
Chapter 2: Problem 29
Find an equation for each line. Write your final answer in slope–interceptform. Perpendicular to \(x+y=18 ; y\) -intercept \((0,-32)\)
Short Answer
Expert verified
The equation is \(y = x - 32\).
Step by step solution
01
Understand the given line
The given line is represented by the equation: \(x + y = 18\). Rewrite it in slope-intercept form (y = mx + b) to identify its slope.
02
Convert to slope-intercept form
Subtract \(x\) from both sides: \(y = -x + 18\). Thus, the slope (m) of this line is \(-1\).
03
Find the slope of the perpendicular line
The slope of any line perpendicular to another is the negative reciprocal of the original line's slope. Therefore, the slope of the perpendicular line is the negative reciprocal of \(-1\), which is \(1\).
04
Use the given y-intercept
The given y-intercept is \((0, -32)\). This means that when \(x = 0\), \(y = -32\). Therefore, the equation will take the form: \(y = mx + b\), where \(m = 1\) and \(b = -32\).
05
Write the final equation
Substitute the slope (m) and y-intercept (b) values into the slope-intercept form (y = mx + b): \(y = 1x - 32\). Thus, the equation of the line perpendicular to \(x + y = 18\) with a y-intercept of \((0, -32)\) is \(y = x - 32\).
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope-Intercept Form
The slope-intercept form is one of the most commonly used ways to express the equation of a straight line. It’s written as:
\[y = mx + b\]
In this form, **y** represents the y-coordinate, **x** represents the x-coordinate, **m** is the slope of the line, and **b** is the y-intercept. The slope **m** indicates how steep the line is. The y-intercept **b** indicates where the line crosses the y-axis.
To convert any linear equation into slope-intercept form, rearrange the equation to solve for **y**. For example, if you start with an equation like **x + y = 18**, you would subtract **x** from both sides to get: \[y = -x + 18\]This shows the line in slope-intercept form, where **m = -1** and **b = 18**.
Negative Reciprocal Slope
To understand lines that are perpendicular to each other, it's essential to grasp the concept of a negative reciprocal slope. Simply put, if two lines are perpendicular, the slope of one line is the negative reciprocal of the slope of the other line.
To find the negative reciprocal of a number:
- First, take the reciprocal (flip the fraction). For instance, the reciprocal of 2 is 1/2.
- Next, change the sign. If your original slope was positive, the negative reciprocal will be negative, and vice versa.
For example, if the slope of the given line is **-1** (as in the initial problem: **x + y = 18** converted to **y = -x + 18**), the slope of a line perpendicular to this would be **1** because the negative reciprocal of **-1** is **1**.
Y-Intercept
The y-intercept is the point where a line crosses the y-axis. This is an important characteristic because it shows the value of **y** when **x = 0**. In the slope-intercept form of a line (\[y = mx + b\]), **b** represents the y-intercept.
For instance, in the problem solution, the y-intercept is given as \[ (0, -32)\]. This means that the line crosses the y-axis at -32. To put it into the equation format, once we know the slope (which we found to be **1** as it’s perpendicular to **-1**), we can plug in the slope and y-intercept to get: \[y = 1x - 32\]With this, we have the equation of the line that is perpendicular to **x + y = 18** and has a y-intercept of **(0, -32)**.
One App. One Place for Learning.
All the tools & learning materials you need for study success - in one app.
Get started for free
![Problem 29 Find an equation for each line. ... [FREE SOLUTION] (3)](https://i0.wp.com/website-cdn.studysmarter.de/sites/14/2023/12/Rectangle-22721-2.png "Problem 29 Find an equation for each line. ... [FREE SOLUTION] (3)")
Most popular questions from this chapter
Recommended explanations on Math Textbooks
Probability and Statistics
Read ExplanationStatistics
Read ExplanationGeometry
Read ExplanationDecision Maths
Read ExplanationPure Maths
Read ExplanationDiscrete Mathematics
Read ExplanationWhat do you think about this solution?
We value your feedback to improve our textbook solutions.
Study anywhere. Anytime. Across all devices.
Sign-up for free
This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. Accept
Privacy & Cookies Policy
Privacy Overview
This website uses cookies to improve your experience while you navigate through the website. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website. These cookies will be stored in your browser only with your consent. You also have the option to opt-out of these cookies. But opting out of some of these cookies may affect your browsing experience.
Always Enabled
Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that ensures basic functionalities and security features of the website. These cookies do not store any personal information.
Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. It is mandatory to procure user consent prior to running these cookies on your website.
![Problem 29 Find an equation for each line. ... [FREE SOLUTION] (2024)](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mP8/h8AAvMB+NzmkbcAAAAASUVORK5CYII=)