/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 79 A line passes through \((24,15)\... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 79

A line passes through \((24,15)\) and \((21,17)\). Write the equation in slope- intercept form of the parallel line that passes through \((-4,3)\).

Short Answer

Expert verified
The equation of the parallel line is \( y = -\frac{2}{3}x + \frac{1}{3} \).

Step by step solution

01

- Find the slope of the original line

The slope (m) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\). Substituting \((x_1, y_1) = (24, 15)\) and \((x_2, y_2) = (21, 17)\), we get: \[ m = \frac{{17 - 15}}{{21 - 24}} = \frac{2}{-3} = -\frac{2}{3} \]
02

- Use the same slope for the parallel line

Since parallel lines have the same slope, the slope of the line we need to find is also \(-\frac{2}{3}\).
03

- Use the point-slope form of the equation

The point-slope form of the equation of a line is \[ y - y_1 = m(x - x_1) \]. We use the point \((-4, 3)\) and slope \(-\frac{2}{3}\). Substituting these values, we get: \[ y - 3 = -\frac{2}{3}(x + 4) \]
04

- Simplify to get the slope-intercept form

Expand and simplify the equation from the previous step to get the slope-intercept form \(y = mx + b\): \[ y - 3 = -\frac{2}{3}(x + 4) \] \[ y - 3 = -\frac{2}{3}x - \frac{8}{3} \] \[ y = -\frac{2}{3}x - \frac{8}{3} + 3 \] \[ y = -\frac{2}{3}x - \frac{8}{3} + \frac{9}{3} \] \[ y = -\frac{2}{3}x + \frac{1}{3} \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope Calculation
The slope of a line tells us how steep the line is. We calculate the slope, often represented as \(m\), by using two points on the line. The formula is: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
In the given problem, the points are \((24, 15)\) and \((21, 17)\).
Substituting these values into the formula, we get:
\( m = \frac{17 - 15}{21 - 24} = \frac{2}{-3} = -\frac{2}{3} \)
This means the slope of the original line is \(-\frac{2}{3}\).
Remember, the slope is a measure of how much y changes for a corresponding change in x.
Parallel Lines
Parallel lines have a special property: they always have identical slopes. This means if one line has a slope of \(m\), any line parallel to it also has slope \(m\).
In our example, the original line's slope is \(-\frac{2}{3}\). Thus, a parallel line must also have a slope of \(-\frac{2}{3}\).
This consistency helps us when finding the equation of a parallel line because we can directly use the same slope in our calculations.
Point-Slope Form
The point-slope form of a linear equation is particularly helpful when you know a point on the line \( (x_1, y_1) \) and the slope \(m\).
The formula is given by:
\( y - y_1 = m(x - x_1) \).
For the line parallel to the one passing through \ ((24, 15)) and ((21, 17)) \, we use the point \ ((-4, 3)) \ and slope \ (-\frac{2}{3}) \.
Substituting these values, we get:
\( y - 3 = -\frac{2}{3}(x + 4) \).
This equation can now be simplified to get the slope-intercept form.
Linear Equations
Linear equations can be represented in slope-intercept form, which is \( y = mx + b \). Here, \(m\) is the slope and \(b\) is the y-intercept.
Starting with the point-slope form \( y - 3 = -\frac{2}{3}(x + 4) \), we simplify:
\[ y - 3 = -\frac{2}{3}x - \frac{8}{3} \]
Adding 3 to both sides, we get:
\[ y = -\frac{2}{3}x - \frac{8}{3} + 3 \]
Since 3 can be written as \( \frac{9}{3} \), we have:
\[ y = -\frac{2}{3}x + \frac{1}{3} \]
Thus, the equation of the parallel line in slope-intercept form is \( y = -\frac{2}{3}x + \frac{1}{3} \).
This form makes it easy to graph the line and identify its slope and y-intercept.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Use the slope formula to find the slope of the line that passes through \((5,-9)\) and \((-2,-11)\). $$ \text { Incorrect Answer: } \begin{aligned} m &=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ m &=\frac{-11-9}{-2-5} \\ m &=\frac{-20}{-7} \\ m &=\frac{20}{7} \end{aligned} $$

An employee in Maine has two jobs that pay minimum wage. He works \(28 \mathrm{hr}\) per week at one job and \(18 \mathrm{hr}\) per week at the other job. Find the difference in his pay per week between October 2007 and October \(2009 .\) October 1, 2007鈥揗inimum Wage is \(\$ 7.00\) per hour October 1, 2008-Minimum Wage is \(\$ 7.25\) per hour October 1, 2009-Minimum Wage is \(\$ 7.50\) per hour (Source: www.maine.gov/labor/posters/minimumwage.pdf)

For exercises 89-92, the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Find the slope of the line that passes through \((7,1)\) and \((9,4)\). Incorrect Answer: \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) $$ \begin{aligned} m &=\frac{9-7}{4-1} \\ m &=\frac{2}{3} \end{aligned} $$

Use the slope formula to find the slope of the line that passes through the points. \((0,-9) ;(-2,0)\)

Some students find it helpful to use their learning preferences as a guide in how to study. Visual Learner \- Take detailed notes during class. Use colored pens and highlighters. \- Reorganize and rewrite notes after class; draw diagrams that summarize what you have learned. \- Read your book; watch the videos or DVDs for this text. \- Use flash cards for memory work. \- Sit where you can see everything in the classroom. Turn your phone or tablet off so that you are not distracted. Auditory Learner \- With permission, record your class. Take only brief notes of the big ideas and examples. After class, listen to the recording. Complete your notes. Restate the main ideas aloud to yourself. Use videos and DVDs to fill in anything you missed in class. \- Talk to yourself as you do your homework. Explain each step to yourself. \- Do memory work by repeating definitions aloud. Listen to a recording of the words and definitions. Create songs that help you remember a definition. \- Sit where you can hear everything. Turn your phone or tablet off so that you are not distracted. Kinesthetic Learner \- With permission, record your class. Take brief notes of the big ideas and examples. After class, listen to the recording. Complete your notes. Draw pictures. Use videos and DVDs to fill in anything you missed during class. -With your finger, trace diagrams and graphs. Do not just look at them. \- Imagine symbols such as variables as three-dimensional objects or even cartoon characters. Imagine yourself counting them, combining them, or subtracting them. \- Do memory work as you exercise or walk to your car. Walk around your room as you repeat definitions. You may find it helpful to come up with physical motions and/or a song that correspond to a procedure. \- If your class is mostly lecture, prepare yourself mentally before you walk into class to concentrate and not daydream. Turn your phone or tablet off so that you are not distracted. Identify any strategies listed that you don't currently use but you think might be helpful.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.