/*! 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 72 Find the slope and the y-interce... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 72

Find the slope and the y-intercept of the graph of each equation and graph it. See Examples 4 and 5. $$ 2 x+3 y=9 $$

Short Answer

Expert verified
The slope is \(-\frac{2}{3}\) and the y-intercept is 3.

Step by step solution

01

Convert to Slope-Intercept Form

First, we need to manipulate the given equation into the slope-intercept form, which is \( y = mx + b \), where \( m \) represents the slope and \( b \) the y-intercept. Start with the equation: \[ 2x + 3y = 9 \] Subtract \( 2x \) from both sides: \[ 3y = -2x + 9 \]Then, divide every term by 3 to solve for \( y \):\[ y = -\frac{2}{3}x + 3 \]
02

Identify the Slope and Y-intercept

From the equation \( y = -\frac{2}{3}x + 3 \), identify the slope \( m \) and the y-intercept \( b \):- The slope \( m \) is \(-\frac{2}{3}\).- The y-intercept \( b \) is 3.
03

Plot the Y-intercept

Start by plotting the y-intercept on the graph, which is the point where \( y = 3 \). This is the point \( (0, 3) \) on the y-axis.
04

Use the Slope to Plot Another Point

The slope \(-\frac{2}{3}\) means that for every 2 units you move down on the y-axis, you move 3 units to the right on the x-axis. From the y-intercept \( (0, 3) \), move down 2 units to \( y = 1 \) and 3 units to the right to \( x = 3 \). This gives you the point \( (3, 1) \) on the graph.
05

Draw the Line

Connect the two points \( (0, 3) \) and \( (3, 1) \) with a straight line. Extend the line in both directions, ensuring it passes through these two points. This line represents the graph of the equation \( 2x + 3y = 9 \).

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-Intercept Form
The slope-intercept form of a linear equation is a powerful and intuitive way to write equations of lines. This form is expressed as \( y = mx + b \). Here are the key components of this format:
  • \( m \) represents the slope of the line, illustrating how steep the line is.
  • \( b \) stands for the y-intercept, the point where the line crosses the y-axis.
Understanding this form is crucial because it provides direct information about the slope and y-intercept just by looking at the equation. When given a linear equation not in this form, like \( 2x + 3y = 9 \), our first task is to rearrange it. By isolating \( y \), we derive \( y = -\frac{2}{3}x + 3 \), clearly showing the slope and y-intercept. This makes it much easier to interpret and graph the equation later.
Graphing Linear Equations
Graphing linear equations can be simplified by using the slope-intercept form, \( y = mx + b \). Once you have identified the slope and y-intercept from a given equation, you can plot it easily on a coordinate grid.
Here鈥檚 a simple guide to graphing:
  • Start by plotting the y-intercept \((0, b)\) on the y-axis. This tells you where to begin your line.
  • Use the slope \( m \), often expressed as a fraction \( \frac{rise}{run} \), to determine the direction and steepness of the line. A negative slope, like \(-\frac{2}{3}\), indicates a line that falls as it moves from left to right.
  • From the y-intercept, apply the slope. For a slope of \(-\frac{2}{3}\), move down 2 units and right 3 units to find your next point.
  • After marking at least two points, draw a straight line through them to represent the entire equation.
Upon completing these steps, you will see how the graph visually represents the equation, showcasing the relationship between \( x \) and \( y \).
Identifying Slope and Y-Intercept
In linear equations, identifying both the slope and y-intercept is a fundamental step in understanding and graphing these equations. These two values tell us a lot about the line:
  • The slope\( (m) \) indicates the tilt or steepness of the line. A positive slope means the line rises from left to right, while a negative slope means it falls.
  • The y-intercept\( (b) \) is where the line crosses the y-axis. It gives us a starting point to begin drawing our line on a graph.
In the equation \( y = -\frac{2}{3}x + 3 \), the slope of \(-\frac{2}{3}\) shows us the pattern of descent the line follows, and the y-intercept of 3 tells us the line crosses the y-axis at this point. By correctly identifying these parts, we gain valuable insights into the line's behavior without needing to look at its graph.

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

Cable TV. A 186 -foot television cable is to be cut into four pieces. Find the length of each piece if each successive piece is 3 feet longer than the previous one.

iPads. When a student purchased an Apple iPad with \(\mathrm{Wi}_{1}-\mathrm{Fi}+3 \mathrm{G}\) for \(\$ 629.99,\) he also enrolled in a \(250 \mathrm{MB}\) data plan that cost \(\$ 14.95\) per month. a. Write a linear equation that gives the cost for him to purchase and use the iPad for \(m\) months. b. Use your answer to part a to find the cost to purchase and use the iPad for 2 years.

Why is \((0,0)\) not an acceptable choice for a test point when graphing a linear inequality whose boundary passes through the origin?

A line having slope \(\frac{2}{3}\) passes through the point \((10,-12)\) What is the \(y\) -coordinate of another point on the line whose \(x\) -coordinate is \(16 ?\)

Find the slope and the y-intercept of the graph of each equation and graph it. See Examples 4 and 5. $$ 2 x+y=-6 $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Geometry

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.