1. Splash Screen

2. Five-Minute Check (over Chapter 2) CCSS Then/Now New Vocabulary
Example 1: Solve by Using a Table
Example 2: Solve by Graphing
Example 3: Classify Systems
Concept Summary: Characteristics of Linear Systems
Key Concept: Substitution Method
Example 4: Real-World Example: Use the Substitution Method
Key Concept: Elimination Method
Example 5: Solve by Using Elimination
Example 6: Standardized Test Example: No Solution and Infinite Solutions
Concept Summary: Solving Systems of Equations
Lesson Menu

3. Find the domain and range of the relation {(–4, 1), (0, 0), (1, –4), (2, 0), (–2, 0)}. Determine whether the relation is a function.
A. D = {–4, –2, 0, 1, 2}, R = {–4, 0,1}; yes
B. D = {0, 1, 2}, R = {0, 1}; yes
C. D = {–4, 0, 1}, R = {–4, –2, 0, 1, 2}; no
D. D = {–2, –4}; R = {–4, 0, 1}; yes
5-Minute Check 1

4. Find the domain and range of the relation {(–4, 1), (0, 0), (1, –4), (2, 0), (–2, 0)}. Determine whether the relation is a function.
A. D = {–4, –2, 0, 1, 2}, R = {–4, 0,1}; yes
B. D = {0, 1, 2}, R = {0, 1}; yes
C. D = {–4, 0, 1}, R = {–4, –2, 0, 1, 2}; no
D. D = {–2, –4}; R = {–4, 0, 1}; yes
5-Minute Check 1

5. Find the value of f(4) for f(x) = 8 – x – x2.
B. 12
C. –12
D. –16
5-Minute Check 2

6. Find the value of f(4) for f(x) = 8 – x – x2.
B. 12
C. –12
D. –16
5-Minute Check 2

7. Find the slope of the line that passes through (5, 7) and (–1, 0).B.
C. 2
D. 7
5-Minute Check 3

8. Find the slope of the line that passes through (5, 7) and (–1, 0).B.
C. 2
D. 7
5-Minute Check 3

9. Write an equation in slope-intercept form for the line that has x-intercept –3 and y-intercept 6.
A. y = –3x + 6
B. y = –3x – 6
C. y = 3x + 6
D. y = 2x + 6
5-Minute Check 4

10. Write an equation in slope-intercept form for the line that has x-intercept –3 and y-intercept 6.
A. y = –3x + 6
B. y = –3x – 6
C. y = 3x + 6
D. y = 2x + 6
5-Minute Check 4

11. The Math Club is using the prediction equation y = 1
The Math Club is using the prediction equation y = 1.25x + 10 to estimate the number of members it will have, where x represents the number of years the club has been in existence. About how many members does the club expect to have in its fifth year?
A. 15
B. 16
C. 17
D. 18
5-Minute Check 5

12. The Math Club is using the prediction equation y = 1
The Math Club is using the prediction equation y = 1.25x + 10 to estimate the number of members it will have, where x represents the number of years the club has been in existence. About how many members does the club expect to have in its fifth year?
A. 15
B. 16
C. 17
D. 18
5-Minute Check 5

13. Identify the type of function represented by the equation y = 4x2 + 6.
A. absolute value
B. linear
C. piecewise-defined
D. quadratic
5-Minute Check 6

14. Identify the type of function represented by the equation y = 4x2 + 6.
A. absolute value
B. linear
C. piecewise-defined
D. quadratic
5-Minute Check 6

15. Mathematical Practices 2 Reason abstractly and quantitatively.
6 Attend to precision.
CCSS

16. You graphed and solved linear equations.
Solve systems of linear equations graphically.
Solve systems of linear equations algebraically.
Then/Now

17. break-even point system of equations consistent inconsistent
independent
dependent
substitution method
elimination method
Vocabulary

18. Solve by Using a Table
Solve the system of equations by completing a table. x + y = 3 –2x + y = –6
Solve for y in each equation.
x + y = 3
y = –x + 3
–2x + y = –6
y = 2x – 6
Example 1

19. Use a table to find the solution that satisfies both equations.
Solve by Using a Table
Use a table to find the solution that satisfies both equations.
Answer:
Example 1

20. Use a table to find the solution that satisfies both equations.
Solve by Using a Table
Use a table to find the solution that satisfies both equations.
Answer: The solution to the system is (3, 0).
Example 1

21. What is the solution of the system of equations? x + y = 2 x – 3y = –6
B. (0, 2)
C. (2, 0)
D. (–4, 6)
Example 1

22. What is the solution of the system of equations? x + y = 2 x – 3y = –6
B. (0, 2)
C. (2, 0)
D. (–4, 6)
Example 1

23. Solve the system of equations by graphing. x – 2y = 0 x + y = 6
Solve by Graphing
Solve the system of equations by graphing. x – 2y = 0 x + y = 6
Write each equation in slope-intercept form.
The graphs appear to intersect at (4, 2).
Example 2

24. Check Substitute the coordinates into each equation.
Solve by Graphing
Check Substitute the coordinates into each equation.
x – 2y = 0 x + y = 6 Original equations
4 – 2(2) = = 6 Replace x with and y with 2. ?
0 = 0 6 = 6 Simplify.
Answer:
Example 2

25. Check Substitute the coordinates into each equation.
Solve by Graphing
Check Substitute the coordinates into each equation.
x – 2y = 0 x + y = 6 Original equations
4 – 2(2) = = 6 Replace x with and y with 2. ?
0 = 0 6 = 6 Simplify.
Answer: The solution of the system is (4, 2).
Example 2

26. Which graph shows the solution to the system of equations below? x + 3y = 7 x – y = 3D.
Example 2

27. Which graph shows the solution to the system of equations below? x + 3y = 7 x – y = 3D.
Example 2

28. Write each equation in slope-intercept form.
Classify Systems
A. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. x – y = 5 x + 2y = –4
Write each equation in slope-intercept form.
Example 3

29. Classify Systems
Answer:
Example 3

30. Classify Systems
Answer:
The graphs of the equations intersect at (2, –3). Since there is one solution to this system, this system is consistent and independent.
Example 3

31. Write each equation in slope-intercept form.
Classify Systems
B. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. 9x – 6y = –6 6x – 4y = –4
Write each equation in slope-intercept form.
Since the equations are equivalent, their graphs are the same line.
Example 3

32. Classify Systems
Answer:
Example 3

33. Classify Systems
Answer:
Any ordered pair representing a point on that line will satisfy both equations. So, there are infinitely many solutions. This system is consistent and dependent.
Example 3

34. Write each equation in slope-intercept form.
Classify Systems
C. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. 15x – 6y = 0 5x – 2y = 10
Write each equation in slope-intercept form.
Example 3

35. Classify Systems
Answer:
Example 3

36. Classify Systems
Answer:
The lines do not intersect. Their graphs are parallel lines. So, there are no solutions that satisfy both equations. This system is inconsistent.
Example 3

37. Classify Systems
D. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. f(x) = –0.5x + 2 g(x) = –0.5x + 2 h(x) = 0.5x + 2
Example 3

38. Classify Systems
Answer:
Example 3

39. Classify Systems
Answer:
f(x) and g(x) are consistent and dependent. f(x) and h(x) are consistent and independent. g(x) and h(x) are consistent and independent.
Example 3

40. A. consistent and independent B. consistent and dependent
A. Graph the system of equations below. What type of system of equations is shown? x + y = 5 2x = y – 5
A. consistent and independent
B. consistent and dependent
C. consistent
D. none of the above
Example 3

41. A. consistent and independent B. consistent and dependent
A. Graph the system of equations below. What type of system of equations is shown? x + y = 5 2x = y – 5
A. consistent and independent
B. consistent and dependent
C. consistent
D. none of the above
Example 3

42. A. consistent and independent B. consistent and dependent
B. Graph the system of equations below. What type of system of equations is shown? x + y = 3 2x = –2y + 6
A. consistent and independent
B. consistent and dependent
C. inconsistent
D. none of the above
Example 3

43. A. consistent and independent B. consistent and dependent
B. Graph the system of equations below. What type of system of equations is shown? x + y = 3 2x = –2y + 6
A. consistent and independent
B. consistent and dependent
C. inconsistent
D. none of the above
Example 3

44. A. consistent and independent B. consistent and dependent
C. Graph the system of equations below. What type of system of equations is shown? y = 3x + 2 –6x + 2y = 10
A. consistent and independent
B. consistent and dependent
C. inconsistent
D. none of the above
Example 3

45. A. consistent and independent B. consistent and dependent
C. Graph the system of equations below. What type of system of equations is shown? y = 3x + 2 –6x + 2y = 10
A. consistent and independent
B. consistent and dependent
C. inconsistent
D. none of the above
Example 3

46. A. f(x) and g(x) are consistent and dependent.
D. Graph the system of equations below. Which statement is not true? f(x) = x g(x) = x + 4
A. f(x) and g(x) are consistent and dependent.
B. f(x) and g(x) are inconsistent.
C. f(x) and h(x) are consistent and independent.
D. g(x) and h(x) are consistent.
Example 3

47. A. f(x) and g(x) are consistent and dependent.
D. Graph the system of equations below. Which statement is not true? f(x) = x g(x) = x + 4
A. f(x) and g(x) are consistent and dependent.
B. f(x) and g(x) are inconsistent.
C. f(x) and h(x) are consistent and independent.
D. g(x) and h(x) are consistent.
Example 3

48. Concept

49. Concept

50. You are asked to find the number of each type of chair sold.
Use the Substitution Method
FURNITURE Lancaster Woodworkers Furniture Store builds two types of wooden outdoor chairs. A rocking chair sells for $265 and an Adirondack chair with footstool sells for $320. The books show that last month, the business earned $13,930 for the 48 outdoor chairs sold. How many of each chair were sold?
Understand
You are asked to find the number of each type of chair sold.
Example 4

51. x + y = 48 The total number of chairs sold was 48.
Use the Substitution Method
Plan
Define variables and write the system of equations. Let x represent the number of rocking chairs sold and y represent the number of Adirondack chairs sold.
x + y = 48 The total number of chairs sold was 48.
265x + 320y = 13,930 The total amount earned was $13,930.
Example 4

52. x = 48 – y Subtract y from each side.
Use the Substitution Method
Solve one of the equations for one of the variables in terms of the other. Since the coefficient of x is 1, solve the first equation for x in terms of y.
x + y = 48 First equation
x = 48 – y Subtract y from each side.
Example 4

53. Solve Substitute 48 – y for x in the second equation.
Use the Substitution Method
Solve Substitute 48 – y for x in the second equation.
265x + 320y = 13,930 Second equation
265(48 – y) + 320y = 13,930 Substitute 48 – y for x.
12,720 – 265y + 320y = 13,930 Distributive Property
55y = 1210 Simplify.
y = 22 Divide each side by 55.
Example 4

54. x = 26 Subtract 22 from each side.
Use the Substitution Method
Now find the value of x. Substitute the value for y into either equation.
x + y = 48 First equation
x = 48 Replace y with 22.
x = 26 Subtract 22 from each side.
Answer:
Example 4

55. x = 26 Subtract 22 from each side.
Use the Substitution Method
Now find the value of x. Substitute the value for y into either equation.
x + y = 48 First equation
x = 48 Replace y with 22.
x = 26 Subtract 22 from each side.
Answer: They sold 26 rocking chairs and 22 Adirondack chairs.
Example 4

56. Check You can use a graphing calculator to check this solution.
Use the Substitution Method
Check You can use a graphing calculator to check this solution.
Example 4

57. AMUSEMENT PARKS At Amy’s Amusement Park, tickets sell for $24
AMUSEMENT PARKS At Amy’s Amusement Park, tickets sell for $24.50 for adults and $16.50 for children. On Sunday, the amusement park made $6405 from selling 330 tickets. How many of each kind of ticket was sold?
A. 210 adult; 120 children
B. 120 adult; 210 children
C. 300 children; 30 adult
D. 300 children; 30 adult
Example 4

58. AMUSEMENT PARKS At Amy’s Amusement Park, tickets sell for $24
AMUSEMENT PARKS At Amy’s Amusement Park, tickets sell for $24.50 for adults and $16.50 for children. On Sunday, the amusement park made $6405 from selling 330 tickets. How many of each kind of ticket was sold?
A. 210 adult; 120 children
B. 120 adult; 210 children
C. 300 children; 30 adult
D. 300 children; 30 adult
Example 4

59. Concept

60. Use the elimination method to solve the system of equations.
Solve by Using Elimination
Use the elimination method to solve the system of equations.
x + 2y = 10 x + y = 6
In each equation, the coefficient of x is 1. If one equation is subtracted from the other, the variable x will be eliminated.
x + 2y = 10
(–)x + y = 6
y = 4 Subtract the equations.
Example 5

61. Now find x by substituting 4 for y in either original equation.
Solve by Using Elimination
Now find x by substituting 4 for y in either original equation.
x + y = 6 Second equation
x + 4 = 6 Replace y with 4.
x = 2 Subtract 4 from each side.
Answer:
Example 5

62. Now find x by substituting 4 for y in either original equation.
Solve by Using Elimination
Now find x by substituting 4 for y in either original equation.
x + y = 6 Second equation
x + 4 = 6 Replace y with 4.
x = 2 Subtract 4 from each side.
Answer: The solution is (2, 4).
Example 5

63. Use the elimination method to solve the system of equations
Use the elimination method to solve the system of equations. What is the solution to the system? x + 3y = 5 x + 5y = –3
A. (2, –1)
B. (17, –4)
C. (2, 1)
D. no solution
Example 5

64. Use the elimination method to solve the system of equations
Use the elimination method to solve the system of equations. What is the solution to the system? x + 3y = 5 x + 5y = –3
A. (2, –1)
B. (17, –4)
C. (2, 1)
D. no solution
Example 5

65. Solve the system of equations. 2x + 3y = 12 5x – 2y = 11
No Solution and Infinite Solutions
Solve the system of equations. 2x + 3y = 12 5x – 2y = 11
A. (2, 3)
B. (6, 0)
C. (0, 5.5)
D. (3, 2)
Read the Test Item You are given a system of two linear equations and are asked to find the solution.
Example 6

66. No Solution and Infinite Solutions
Solve the Test Item Multiply the first equation by 2 and the second equation by 3. Then add the equations to eliminate the y variable.
2x + 3y = 12 4x + 6y = 24
Multiply by 2.
Multiply by 3.
5x – 2y = 11 (+)15x – 6y = 33
19x = 57
x = 3
Example 6

67. Replace x with 3 and solve for y.
No Solution and Infinite Solutions
Replace x with 3 and solve for y.
2x + 3y = 12 First equation
2(3) + 3y = 12 Replace x with 3.
6 + 3y = 12 Multiply.
3y = 6 Subtract 6 from each side.
y = 2 Divide each side by 3.
Answer:
Example 6

68. Replace x with 3 and solve for y.
No Solution and Infinite Solutions
Replace x with 3 and solve for y.
2x + 3y = 12 First equation
2(3) + 3y = 12 Replace x with 3.
6 + 3y = 12 Multiply.
3y = 6 Subtract 6 from each side.
y = 2 Divide each side by 3.
Answer: The solution is (3, 2). The correct answer is D.
Example 6

69. Solve the system of equations. x + 3y = 7 2x + 5y = 10
B. (1, 2)
C. (–5, 4)
D. no solution
Example 6

70. Solve the system of equations. x + 3y = 7 2x + 5y = 10
B. (1, 2)
C. (–5, 4)
D. no solution
Example 6

71. Concept

72. End of the Lesson