/*! 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 22 In Exercises \(7-24,\) find the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 22

In Exercises \(7-24,\) find the general form of the equation of the line satisfying the conditions given and graph the line. Through \(\left(2, \frac{1}{2}\right)\) and \((-3,8)\)

Short Answer

Expert verified
The general form of the line equation is \(3x + 2y - 7 = 0\).

Step by step solution

01

- Calculate the Slope

Use the slope formula, which is \(m = \frac{y_2 - y_1}{x_2 - x_1}\), where \(x_1, y_1\) are the coordinates of the first point and \(x_2, y_2\) are the coordinates of the second point. Substitute \(x_1 = 2, y_1 = \frac{1}{2}\) and \(x_2 = -3, y_2 = 8\) into the formula: \[m = \frac{8 - \frac{1}{2}}{-3 - 2} = \frac{\frac{16}{2} - \frac{1}{2}}{-5} = \frac{\frac{15}{2}}{-5} = -\frac{3}{2}\]
02

- Use Point-Slope Form

The point-slope form of the equation of a line is \(y - y_1 = m(x - x_1)\). Using point \(2, \frac{1}{2}\) and slope \(-\frac{3}{2}\), substitute these values into the formula: \[y - \frac{1}{2} = -\frac{3}{2}(x - 2)\]
03

- Simplify the Equation

Distribute the slope on the right-hand side: \[y - \frac{1}{2} = -\frac{3}{2}x + 3\] Add \(\frac{1}{2}\) to both sides to get the equation in slope-intercept form: \[y = -\frac{3}{2}x + 3 + \frac{1}{2} = -\frac{3}{2}x + \frac{7}{2}\]
04

- Convert to General Form

To convert the equation to the general form \(Ax + By + C = 0\), multiply both sides by 2 to eliminate the fraction: \[2y = -3x + 7\] Then, rearrange all terms to one side: \[3x + 2y - 7 = 0\]
05

- Graph the Line

To graph the line, plot the points \(2, \frac{1}{2}\) and \(-3, 8\) on the coordinate plane. Draw a straight line that passes through these points.

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 first step in finding the equation of a line is to calculate the slope. The slope measures how steep the line is and is calculated using the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). This formula finds the change in the y-coordinates divided by the change in the x-coordinates.
Given the points \((2, \frac{1}{2})\) and \((-3,8)\), we substitute these coordinates into the formula:
\(m = \frac{8 - \frac{1}{2}}{-3 - 2} = \frac{\frac{16}{2} - \frac{1}{2}}{-5} = \frac{\frac{15}{2}}{-5} = -\frac{3}{2}\)
Here, the slope \(m\) is \(-\frac{3}{2}\).
Point-Slope Form
Once you have the slope, the next step is to use the point-slope form of a line equation, which is: \(y - y_1 = m(x - x_1)\). This form is particularly useful because it only requires the slope \(m\) and one point on the line \((x_1, y_1)\).
Substituting the point \( (2, \frac{1}{2}) \) and slope \( -\frac{3}{2} \) into the formula, we get:
\(y - \frac{1}{2} = -\frac{3}{2}(x - 2)\)
Slope-Intercept Form
The slope-intercept form is a more familiar equation type, written as \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. To convert our current equation \(y - \frac{1}{2} = -\frac{3}{2}(x - 2)\) into slope-intercept form, we need to simplify it.
First, distribute the slope on the right side:
\(y - \frac{1}{2} = -\frac{3}{2}x + 3\)
Then, add \(\frac{1}{2}\) to both sides:
\(y = -\frac{3}{2}x + 3 + \frac{1}{2} = -\frac{3}{2}x + \frac{7}{2}\)
Now, the equation is in slope-intercept form.
Graphing Lines
Graphing a line involves plotting points and drawing the line through them. For the equation \(3x + 2y - 7 = 0\), we can start by plotting the points \( (2, \frac{1}{2}) \) and \((-3,8)\) since these points were given and used to find the equation.
To graph:
  • Plot the point \((2, \frac{1}{2})\).
  • Plot the point \((-3,8)\).
  • Draw a straight line through these points extending in both directions.

The line will show all points that satisfy the equation. Double-checking these points on paper will help confirm your graph is correct.

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

In Exercises \(7-24,\) find the general form of the equation of the line satisfying the conditions given and graph the line. Through \((4,-1)\) and \(\left(7, \frac{2}{3}\right)\)

Determine the intercepts and graph each linear equation. $$x+y-2=0$$

In Exercises \(7-24,\) find the general form of the equation of the line satisfying the conditions given and graph the line. With \(x\) -intercept \((4,0)\) and \(y\) -intercept \((0,-3)\)

In Exercises \(33-38\), determine whether the lines with the given equations are parallel, perpendicular, or neither. \(4 x-y+7=0\) \(3 x-1=0\)

Find the missing coordinate so that each ordered pair is a solution to the equation. $$3 y+1=0$$ (a) \((0, ?) ;\) (b) \((?, 0) ;\) (c) \((3, ?) ; \quad\) (d) \((?,-1)\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

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.