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

(Please note that browsers Safari and Google Chrome do not support all animations and

therefore users may not be able to view the animations on these browsers.)

Page 135

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

(Please note that browsers Safari and Google Chrome do not support all animations and therefore

users may not be able to view the animations on these browsers.)

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)

Page 136

3. (−4, −2)

Animation: Graphing the Solution Set to a Linear Equation in Two Variables

(Please note that browsers Safari and Google Chrome do not support all

animations and therefore users may not be able to view the animations on these

browsers.)

Concept 3: Graphing Linear Equations in Two Variables

Animation: Graphing the Solution Set to a Linear Equation in Two Variables

(Please note that browsers Safari and Google Chrome do not support all animations and therefore

users may not be able to view the animations on these browsers.)

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

16.

16. 4x + 3y = 6

x y

2

3

−1

17.

17.

x y

0

5

−5

18.

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

22. 5x + 3y = 15

23. y = −3x + 5

24. y = −2x + 2

Page 137

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

34. 2x − 5y = 10

35. x − 2y = 4

36. x + y = 8

37. 5x = 3y

38. 3y = −5x

Page 138

39. y = 2x + 4

40. y = −3x − 1

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

Page 139

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

Page 140

53. 2x + 6 = 5

54. −3x = 12

55. −2y + 1 = 9

Exercise: Graphing a Horizontal Line

PDF Transcript for Exercise: Graphing a Horizontal Line

56. −5y = −10