/*! 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 37 (a) find the y-intercept. (b) ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 37

(a) find the y-intercept. (b) find the x-intercept. (c) find a third solution of the equation. (d) graph the equation. \(7 x+2 y=28\)

Short Answer

Expert verified
Intercepts are (0, 14) and (4, 0). A third solution is (2, 7). Plot and draw the line through these points.

Step by step solution

01

- Find the y-Intercept

To find the y-intercept, set x = 0 in the equation and solve for y. Start with the equation: \[ 7x + 2y = 28 \] Set \( x = 0 \): \[ 7(0) + 2y = 28 \] This simplifies to \( 2y = 28 \). Divide both sides by 2: \[ y = 14 \] Thus, the y-intercept is at (0, 14).
02

- Find the x-Intercept

To find the x-intercept, set y = 0 in the equation and solve for x. Start with the equation: \[ 7x + 2y = 28 \] Set \( y = 0 \): \[ 7x + 2(0) = 28 \] This simplifies to \( 7x = 28 \). Divide both sides by 7: \[ x = 4 \] Thus, the x-intercept is at (4, 0).
03

- Find a Third Solution

To find a third solution, choose any value for x or y and solve for the other variable. Choose \( x = 2 \). Substitute x into the equation: \[ 7(2) + 2y = 28 \] This simplifies to \( 14 + 2y = 28 \). Subtract 14 from both sides: \[ 2y = 14 \]. Divide by 2: \[ y = 7 \] Thus, (2, 7) is a third solution.
04

- Graph the Equation

Plot the intercepts (0, 14) and (4, 0) on a coordinate plane. Then plot the third point (2, 7). Draw a straight line through all three points. This line is the graph of the equation \( 7x + 2y = 28 \).

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.

Understanding the y-Intercept
The y-intercept of a linear equation is the point at which the graph of the equation crosses the y-axis. To find the y-intercept, set the value of x to 0 and solve for y. For instance, in the equation \(7x + 2y = 28\), substituting \(x = 0\) simplifies to \(2y = 28\). Solving this equation yields \(y = 14\). Hence, the y-intercept is the point (0, 14). This point indicates where the line touches the y-axis.

Key points to remember:
  • Set \(x = 0\) to find the y-intercept.
  • Solving for y gives the y-coordinate of the intercept.
  • Write the y-intercept as a point (0, y).
Understanding the x-Intercept
The x-intercept of a linear equation is the point where the graph touches the x-axis. To find the x-intercept, set the value of y to 0 and solve for x. For example, given the equation \(7x + 2y = 28\), setting \(y = 0\) simplifies to \(7x = 28\). Solving for x, we find \(x = 4\). So, the x-intercept is at (4, 0). This tells us where the graph will cross the x-axis.

Important points to note:
  • Set \(y = 0\) to find the x-intercept.
  • Solving for x gives the x-coordinate of the intercept.
  • Represent the x-intercept as a point (x, 0).
Graphing Linear Equations
Graphing linear equations involves plotting points and drawing a line through them. Begin by finding the intercepts. In our case, we have the y-intercept (0, 14) and the x-intercept (4, 0). Next, find a third point to ensure accuracy, such as (2, 7) from substituting \(x = 2\) into the equation \(7x + 2y = 28\). After plotting these points on a coordinate plane, draw a straight line through them. This line represents the solution set of the equation.

Steps for graphing:
  • Find and plot the y-intercept.
  • Find and plot the x-intercept.
  • Determine a third point and plot it.
  • Draw a line through all plotted points.
Finding Additional Solutions
Finding extra solutions of a linear equation can be done easily by choosing any value for x or y, then solving for the other variable. For the equation \(7x + 2y = 28\), we found the third solution by setting \(x = 2\). Substituting x in the equation, we get \(7(2) + 2y = 28\), which simplifies to \(14 + 2y = 28\). Solving for y, we get \(y = 7\). Thus, (2, 7) is an additional solution.

Remember:
  • Select a value for x or y.
  • Substitute this value in the equation.
  • Solve for the remaining variable.
  • Write down the solution as a point (x, y).

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

According to the Pew Research Center, the Millennial Generation in the United States includes about 50 million people born after 1980 . Find the number of this generation who say that "you can't be too careful" when dealing with people. Round to the nearest tenth of a million. Whether as a by-product of protective parents, the age of terrorism or a media culture that focuses on dangers, they [Millennials] cast a wary eye on human nature. Two-thirds say "you can't be too careful" when dealing with people. (Source: www.pewresearch.org/millennials, 2010)

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)

(a) represent the information as two ordered pairs. (b) find the average rate of change, \(m\). The estimated 12-month total service revenues for wireless service in the United States increased from \(\$ 113,538,221\) in 2005 to \(\$ 159,929,648\) in 2010 . Round to the nearest thousand. (Source: www.ctia.org)

For exercises 67-78, (a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(4 x+9 y=72\)

For exercises 103-104, some learning preferences describe how you prefer to receive, think about, and learn new information. These preferences include visual learning, auditory learning, and kinesthetic learning. Many students use more than one of these categories as they learn mathematics. \- Visual learners prefer to see information. Although you definitely listen to your instructor, you also like to see the example on a white board or screen. You may be able to recall a process by visualizing it in your mind; you may learn better by organizing information in charts, tables, diagrams, or pictures. You may prefer the use of colored markers instead of just black. \- Auditory learners prefer to hear information. Although you definitely watch what your instructor is doing, you also like your instructor to explain things aloud as he or she works. You may find it difficult to take notes because you cannot concentrate enough on what is being said while you write. You may learn better if you have the chance to explain things to others. \- Kinesthetic learners prefer to do. You may find it difficult to sit still and just watch and listen; you want to be trying it out. You may find that you must take notes in order to learn. If you only watch and listen, you may understand the concept but not remember it after you leave the classroom. You often learn better if you can show others how to do things. Do you have a strong preference for visual, auditory, or kinesthetic learning?
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Discrete 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.