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Chapter 3: Problem 74

(a) complete the table of solutions. (b) graph the equation. $$ \begin{aligned} &y=x+3\\\ &\begin{array}{|l|l|} \hline x & y \\ \hline 0 & \\ \hline 2 & \\ \hline 4 & \\ \hline \end{array} \end{aligned} $$

Short Answer

Expert verified
Table: x=0, y=3; x=2, y=5; x=4, y=7. Graph: Plot points and draw the line y=x+3.

Step by step solution

01

- Identify the equation

Given the equation: y = x + 3
02

- Substitute x values into the equation

Substitute the given x-values from the table into the equation to find the corresponding y-values.
03

- Calculate y for x=0

When x=0: y = 0 + 3 y = 3. Fill this in the table.
04

- Calculate y for x=2

When x=2: y = 2 + 3 y = 5. Fill this in the table.
05

- Calculate y for x=4

When x=4: y = 4 + 3 y = 7. Fill this in the table.
06

- Complete the table

The completed table of solutions is: | x | y | 0 | 3 | 2 | 5 | 4 | 7 |
07

- Graph the equation

Plot the points from the table (0, 3), (2, 5), and (4, 7) on a graph. Draw a straight line passing through these points to represent the equation y = x + 3.

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Key Concepts

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

Table of Solutions
To solve a problem or understand an equation, creating a table of solutions can be very helpful. This involves choosing various values for one variable (typically x) and calculating the corresponding value of the other variable (y) based on the given equation. For example, given the equation \( y = x + 3 \), if you have chosen the x-values (0, 2, 4), you substitute these values in the equation to find y:
  • When x=0, \( y = 0 + 3 = 3 \)
  • When x=2, \( y = 2 + 3 = 5 \)
  • When x=4, \( y = 4 + 3 = 7 \)

By applying these calculations, we create a table of solutions:
  • (0, 3)
  • (2, 5)
  • (4, 7)
This table makes it clear how values of x and y are related in the equation and is essential for the next step, which is graphing.
Graphing Equations
Graphing equations involves plotting points on a coordinate plane based on the solutions obtained from the table of solutions. Each point corresponds to a pair of \( x \) and \( y \) values. To graph the equation \( y = x + 3 \), use the points from our table of solutions, namely (0, 3), (2, 5), and (4, 7). Follow these steps:
  • Plot the point (0, 3) on the graph, which means 0 units along the x-axis and 3 units up on the y-axis.
  • Plot the point (2, 5) by moving 2 units along the x-axis and 5 units up on the y-axis.
  • Plot the point (4, 7) by moving 4 units along the x-axis and 7 units up on the y-axis.
After plotting these points, draw a straight line passing through them. This line represents the equation \( y = x + 3 \). You can check additional points to ensure accuracy, but if the line passes through all the plotted points, your graph is correct.
Substitution Method
The substitution method is a way to solve equations by replacing one variable with an expression involving the other variable. This method is particularly useful when creating a table of solutions. In our exercise, given the equation \( y = x + 3 \), we substitute specific values for x into the equation to determine the corresponding y-values:
  • When x = 0, substitute 0 for x in the equation: \( y = 0 + 3 = 3 \)
  • When x = 2, substitute 2 for x in the equation: \( y = 2 + 3 = 5 \)
  • When x = 4, substitute 4 for x in the equation: \( y = 4 + 3 = 7 \)
This method helps us to fill the table of solutions efficiently, providing insight into the relationship between the variables in the equation.
Coordinate Plane
The coordinate plane is a two-dimensional surface on which points are plotted using pairs of numbers (coordinates). It consists of a horizontal axis (x-axis) and a vertical axis (y-axis), intersecting at the origin (0,0). Using the coordinate plane allows us to visually represent equations and their solutions. For the equation \( y = x + 3 \):
  • Identify the x-axis and y-axis on the graph paper.
  • Label your axes and mark units along each axis.
  • Plot the points (0, 3), (2, 5), and (4, 7) using the coordinates from our table of solutions.
  • Draw a line through these points to represent the equation.
The use of a coordinate plane helps us to see how equations translate into geometric shapes, such as lines, aiding in a better understanding of linear relationships.

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Most popular questions from this chapter

Use the slope formula to find the slope of the line that passes through the points. \((-30,55) ;(-5,80)\)

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(2 x+5 y=15\)

(a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((0,-5) ;(2,0)\)

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 number of traffic fatalities in Kentucky decreased from 985 deaths in 2005 to 760 deaths in 2010 . (Source: www-nrd.nhtsa.dot.gov)
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