2.1 linear equations in two variables concept 1: the...
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2.1 Linear Equations in Two Variables
Concept 1: The Rectangular Coordinate System
2. Let a and b represent nonzero real numbers. Then
1. An ordered pair of the form (0, b) represents a point on which axis?
2. An ordered pair of the form (a, 0) represents a point on which axis?
3. Given the coordinates of a point, explain how to determine in which quadrant the point is
located.
4. What is meant by the word ordered in the term ordered pair?
Animation: Plotting Points in the Rectangular Coordinate System
Animation: Practicing Plotting Points in the Rectangular Coordinate System
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5. Plot the points on a rectangular coordinate system. (See Example 1.)
1. (−2, 1)
2. (0, 4)
3. (0, 0)
4. (−3, 0)
5. 6. (−4.1, −2.7)
6. Plot the points on a rectangular coordinate system.
1. (−2, 5)
2. 3. (4, −3)
4. (0, −2)
5. (2, 2)
6. (−3, −3)
Exercise: Plotting Points
PDF Transcript for Exercise: Plotting Points
7. A point on the x-axis will have what y-coordinate?
8. A point on the y-axis will have what x-coordinate?
For Exercises 9 and 10, give the coordinates of the labeled points, and state the quadrant or axis
where the point is located.
Animation: Estimating the Coordinates of Points on a Graph
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9.
10.
Exercise: Finding the Coordinates of Points
PDF Transcript for Exercise: Finding the Coordinates of Points
Concept 2: Linear Equations in Two Variables
For Exercises 11–14, determine if the ordered pair is a solution to the linear equation. (See
Example 2.)
11. 2x − 3y = 9
1. (0, −3)
2. (−6, 1)
3.
Exercise: Determining Solutions to a Linear Equation
PDF Transcript for Exercise: Determining Solutions to a Linear Equation
12. −5x − 2y = 6
1. (0, 3)
2. 3. (−2, 2)
13. 1. (−1, 0)
2. (2, 3)
3. (−6, 1)
14. 1. (0, −4)
2. (2, −7)
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3. (−4, −2)
Animation: Graphing the Solution Set to a Linear Equation in Two Variables
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Concept 3: Graphing Linear Equations in Two Variables
Animation: Graphing the Solution Set to a Linear Equation in Two Variables
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For Exercises 15–18, complete the table. Then graph the line defined by the points. (See
Examples 3 and 4.)
15. 3x − 2y = 4
x y
0
4
−1
18.
x y
0
3
6
19.
In Exercises 19–30, graph the linear equation by using a table of points. (See Examples 3 and
4.)
19. x + y = 5
20. x + y = −8
21. 3x − 4y = 12
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25.
26.
27. x = −5y − 5
PDF Transcript for Exercise: Graphing a Linear Equation
Exercise: Graphing a Linear Equation
28. x = 4y + 2
29. x = 2y
30. x = −3y
Concept 4: x- and y-Intercepts
31. Given a linear equation, how do you find an x-intercept? How do you find a y-intercept?
32. Can the point (4, −1) be an x- or y-intercept? Why or why not?
For Exercises 33–44, a. find the x-intercept, b. find the y-intercept, and c. graph the equation.
(See Examples 5 and 6.)
33. 2x + 3y = 18
41.
PDF Transcript for Exercise: Finding the x- and y- intercepts from an Equation
Exercise: Finding the x- and y- intercepts from an Equation
42.
43.
44.
45. A salesperson makes a base salary of $10,000 a year plus a 5% commission on the total
sales for the year. The yearly salary can be expressed as a linear equation as
where y represents the yearly salary and x represents the total yearly sales. (See Example
7.)
1. What is the salesperson's salary for a year in which his sales total $500,000?
2. What is the salesperson's salary for a year in which his sales total $300,000?
3. What does the y-intercept mean in the context of this problem?
4. Why is it unreasonable to use negative values for x in this equation?
46. A taxi company in Miami charges $2.00 for any distance up to the first mile and $1.10
for every mile thereafter. The cost of a cab ride can be modeled graphically.
1. Explain why the first part of the model is represented by a horizontal line.
2. What does the y-intercept mean in the context of this problem?
3. Explain why the line representing the cost of traveling more than 1 mi is not
horizontal.
4. How much would it cost to take a cab
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47. A business owner buys several new computers for the office for $1500 each. The
accounting office depreciates each computer by $300 per year. The value y (in $) for each
computer can be represented by y = 1500 − 300x, where x is the number of years after
purchase.
1. How much will a computer be worth 1 yr after purchase?
2. After how many years will the computer be worth only $300?
3. Determine the y-intercept and interpret its meaning in the context of this problem.
4. Determine the x-intercept and interpret its meaning in the context of this problem.
48. The equation y = −3.6x + 59 can be used to approximate the air temperature y (in °F) at
an altitude x (in 1000 ft).
1. Determine the air temperature at 10,000 ft.
2. At what altitude is the air temperature −5.8°F?
3. Determine the y-intercept and interpret its meaning in the context of this problem.
4. Determine the x-intercept and interpret its meaning in the context of this problem.
Concept 5: Horizontal and Vertical Lines
For Exercises 49–56, identify the line as either vertical or horizontal, and graph the line. (See
Examples 8 and 9.)
49. y = −1
50. y = 3
51. x = 2
Exercise: Graphing a Vertical Line
PDF Transcript for Exercise: Graphing a Vertical Line
52. x = −5
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53. 2x + 6 = 5
54. −3x = 12