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Chapter 3: Problem 38

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

Short Answer

Expert verified
For 2x + 7y = 28: y-intercept is (0, 4); x-intercept is (14, 0); another solution is (7, 2); draw a line through (0, 4), (14, 0), and (7, 2).

Step by step solution

01

Find the y-intercept

To find the y-intercept, set x = 0 in the equation 2x + 7y = 28 and solve for y: 2(0) + 7y = 28 7y = 28 y = 4 The y-intercept is (0, 4).
02

Find the x-intercept

To find the x-intercept, set y = 0 in the equation 2x + 7y = 28 and solve for x: 2x + 7(0) = 28 2x = 28 x = 14 The x-intercept is (14, 0).
03

Find a third solution

Choose any value for x (other than 0 or 14) and solve for y. Let's choose x = 7 : 2(7) + 7y = 28 14 + 7y = 28 7y = 14 y = 2 Another solution is (7, 2).
04

Graph the equation

Plot the three points found: (0, 4), (14, 0), and (7, 2) on the coordinate plane. Then, draw a straight line through these points to represent the equation 2x + 7y = 28.

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Key Concepts

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

Finding Intercepts
Solving linear equations often begins with finding the intercepts. Intercepts are where the graph of the equation crosses the x-axis and y-axis. For the given equation \(2x + 7y = 28\) we can find the intercepts as follows:
  • *Y-Intercept*: To find the y-intercept, set \(x = 0\). Substituting \(x = 0\) in the equation gives you \(2(0) + 7y = 28\). Simplifying this, we get \(7y = 28\) and \(y = 4\). Hence, the y-intercept is (0, 4).
  • *X-Intercept*: To find the x-intercept, set \(y = 0\). Substituting \(y = 0\) in the equation gives you \(2x + 7(0) = 28\). Simplifying this, we get \(2x = 28\) and \(x = 14\). Hence, the x-intercept is (14, 0).
Graphing Linear Equations
Graphing linear equations visually represents the solutions of the equation. For the equation \(2x + 7y = 28\), we already have two intercepts: (0, 4) and (14, 0). To graph this equation effectively, follow these steps:
  • Plot the x-intercept (14, 0) on the coordinate plane.
  • Plot the y-intercept (0, 4) on the coordinate plane.
  • Find a third point for accuracy. We chose \(x = 7\). Substituting into the equation gives \(2(7) + 7y = 28\), which simplifies to \(7y = 14\) and \(y = 2\). Thus, another point is (7, 2). Plot this on the graph as well.
  • Draw a line through all three points. This line represents all solutions to the equation \(2x + 7y = 28\).
Graphing helps visualize the relationships between variables and confirm solutions.
Coordinates
Coordinates are essential in plotting points on a graph. Each point on the graph is represented by a coordinate pair \((x, y)\), where \(x\) indicates the position along the horizontal axis and \(y\) indicates the position along the vertical axis. For example:
  • The y-intercept (0, 4) means \(x = 0\) and \(y = 4\). This point is on the y-axis.
  • The x-intercept (14, 0) means \(x = 14\) and \(y = 0\). This point is on the x-axis.
  • Another coordinate (7, 2) means \(x = 7\) and \(y = 2\). This point lies within the first quadrant.
Understanding coordinates is crucial for plotting points accurately and ensuring the correct graph of the equation.

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Most popular questions from this chapter

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(y=-x+3\)

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. Have you noticed anything that your instructor does while teaching that you find helps you remember what has been taught?

(a) represent the information as two ordered pairs. (b) find the average rate of change, \(m\). The estimated number of wireless connections in the United States increased from 207,896,198 connections in 2005 to \(302,859,674\) connections in 2010 . Round to the nearest thousand. (Source: www.ctia.org)

(a) find three solutions of the equation. (b) graph the equation. \(y=x\)

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(x-3 y=27\)
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