1. Chapter 4 Systems of Equations and Problem Solving
How are systems of equations solved?

2. Activation
Review Yesterday’s Warm-up

3. 4-1 Systems of Equations in two Variables
How do you solve a system of equations in two variables graphically?

4. Vocabulary
Systems of equations: two or more equations using the same variables
Linear systems: each equation has two distinct variables to the first degree.

5. Vocabulary Independent system: one solution
Dependent system: many solutions, the same line
Consistent: having at least 1 solution
Inconsistent system: no solution, parallel lines

6. Directions: Solve each equation for y Graph each equation
State the point of intersection

7. Examples:
x – y = 5 and y + 3 = 2x

8. Examples:
3x + y = 5 and 15x + 5y = 2

9. Examples:
y = 2x + 3 and -4x + 2y = 6

10. Examples:
x – 2y + 1 = 0 and x + 4y – 6 =0

11. What limitations do you think are affiliated with this procedure?
Method only works well when the solution is integer values. Tomorrow we will investigate other methods

12. SUMMARIZATION
1. 2.

13. 4-1 Homework PAGE(S): 161 NUMBERS: 2 – 16 even
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15. Activation
Review Yesterday’s Warm-up

16. 4-2A Solving Systems of Equations —SUBSTITUTION
How do you solve a system of equations in two variables by substitution?

17. Substitution: example: 4x + 3y = 4 2x – y = 7
1) LOOK FOR A VARIABLE W/O A COEFFICIENT
2) SOLVE FOR THAT VARIABLE
3) SUBSTITUTE THIS NEW VALUE INTO THE OTHER EQUATION
example:
4x + 3y = 4
2x – y = 7

18. Example:
2y + x = 1
3y – 2x = 12

19. Examples:
5x + 3y = 6
x - y = -1

20. SUMMARIZATION
1. 2 3

21. 4-2a Homework USING SUBSTITUTION PAGE(S): 166 -167 NUMBERS: 1 – 8 all
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22. Activation
Review Yesterday’s warm-up

23. 4-2B and Solving Systems of Equations —LINEAR COMBINATION —ELIMINATION METHOD Consistent and Dependent Systems
How do you solve a system of equations in two variables by linear combinations?
What makes a system dependent, independent, consistent, or inconsistent?

24. REMEMBER Consistent systems—have at least one solution
Inconsistent systems—have no solution
Independent systems—have exactly one solution
Dependent systems—have an infinite number of solutions

25. Combination/Elimination
1)LOOK FOR OR CREATE A SET OF OPPOSITES
TO CREATE: A) USE THE COEFFICIENT OF THE 1ST WITH THE SECOND AND VICE VERSA
B) MAKE SURE THERE WILL BE ONE positive AND ONE negative
2) ADD THE EQUATIONS TOGETHER AND SOLVE
3) SUBSTITUTE IN EITHER EQUATION AND SOLVE FOR THE REMAINING VARIABLE

26. Example:
4x – 2y = 7
x + 2y = 3

27. Example:
4x + 3y = 4
2x - y = 7

28. Example: 3x – 7y = 15 5x + 2y = -4 Vocabulary: consistent solution
One value of x and one value of y

29. Example: 2x - y = 3 -2x + y = -3 Vocabulary: Dependent solution
0 = 0 is TRUE therefore same line

30. Example: 2x - y = 3 -2x + y = 9 Vocabulary: Inconsistent solution
0 = 12 is FALSE therefore parallel lines
Vocabulary: Independent
Having one or fewer solutions

31. Example:
3x + 2y = 7
5x + 7y = 17

32. SUMMARIZATION

33. 4-2B Homework USING LINEAR COMBINATIONS PAGE(S): 166 -167
NUMBERS: 10 – 22 even
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34. Activation
Review Yesterday’s Warm-up

35. 4-3 Using a system of two Equations
How do you translate real life problems into systems of equations?

36. USE ROPES:
Read the problem
Organize your thoughts in a chart
Plan the equations that will work
Evaluate the Solution
Summarize your findings

37. Example:
The sum of the first number and a second number is -42. The first number minus the second is 52. Find the numbers
1st number x
2nd number y
x y = -42
x y = 52

38. Example:
Soybean meal is 16% protein and corn meal is 9% protein. How many pounds of each should be mixed together to get a 350 pound mix that is 12% protein?
Soybean meal x
.16
Corn meal y
.09
350
.12
x y = 350
.16x + .09y = .12 • 350

39. Example:
A total of $1150 was invested part at 12% and part at 11%. The total yield was $ How much was invested at each rate?
12% investment x
.12
11% investment y
.11
1150
133.75
x y = 1150
.12x + .11y =

40. Example:
One day a store sold 45 pens. One kind cost $8.75 the other $ In all, $ was earned. How many of each kind were sold?
Pen 1 x
8.75
Pen 2 y
9.75 45
398.75
x y = 45
8.75x y =

41. SUMMARIZATION
1. 2.

42. 4-3 Homework PAGE(S): 171 -173 NUMBERS: 4 – 24 by 4’s
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43. Activation
Review Yesterday’s Warm-up

44. 4-4 Systems of Equations in three Variables
How do you solve a system of equations in three variables? How is it similar to solving a system in two equations?

45. Find x, y, z
2x + y - z = 5 3x - y + 2z = -1 x - y - z = 0

46. Find x, y, z
2x - y - 3z = 6 3x + 3y - z = 11 4x + 2y - 3 z = 18

47. Find x, y, z
4x + 9y = 8 8x + 6z = -1 6y + 6z = -1

48. Find x, y, z
2x + z = 7 x + 3y + 2z = 5 4x + 2y - 3z = -3

49. SUMMARIZATION
1. 2.

50. 4-4 Homework PAGE(S): 178 - 179 NUMBERS: 4 – 24 by 4’s
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51. Activation
Review Yesterday’s Warm-up

52. 4-5 Using a System of Three Equations
How do you translate word problems into a system of three equations?

53. Example:
The sum of three numbers is The third is 11 less than ten times the second. Twice the first is 7 more than three times the second. Find the numbers.
1st number x
2nd number y
3rd number Z
x + y + z = 105
z = 10y – 11
2x = y

54. Example: Sawmills A, B, C can produce 7400 board feet of lumber per day. A and B together can produce 4700 board feet, while B and C together can produce 5200 board feet. How many board feet can each mill produce?
Mill A A
Mill B B
Mill C C
A + B + C = 7400
A + B = 4700
B + C = 5200

55. SUMMARIZATION
1. 2 3

56. 4-5 Homework PAGE(S): 181 - 182 NUMBERS: 4, 8, 12, 16
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57. Activation
Review Yesterday’s Warm-up

58. 4-6b Other Systems of Equations
How do you solve other systems of equations?

59. Find x, y when x is an integer
y = x2 y = 3x – 2

60. Find x, y when x is an integer
y = x2 – 2 3x – 2y = 2

61. Find x, y when x is an integer
y + 3 = 2x x2 + 2xy = -1

62. Find x, y when x is an integer
y = x3 + 3x y = 3x – 8

63. 4-6b Homework
PAGE(S): worksheet
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64. 4-7 Systems of Inequalities
How do you solve a system of linear inequalities?

65. Vocabulary:
Feasible region: the area of all possible outcomes

66. Directions: Solve each equation for y Graph each equation
Shade each with lines
Shade the intersecting lines a solid color

67. Examples x – 2y < 6 y ≤ -3/2 x + 5 x – 2y < 6 -2y < 6 – x

68. y ≤ -2x + 4 x > -3

69. y < 4 y ≥ |x – 3|

70. 3x + 4y ≥ 12 5x + 6y ≤ 30 1 ≤ x ≤ 3 3x + 4y ≥ 12 4y≥12-3x y ≥ 3 – ¾ x
2 vert lines

71. SUMMARIZATION
1. 2.

72. 4-7 Homework PAGE(S): 192 NUMBERS: 4 – 32 by 4’s
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73. y≤x - 3
x – y ≥ 3
- y ≥ 3 - x
y ≤ x - 3

74. 2y - x ≤ 4
2y ≤ 4 + x
y ≤ /2x
y < x
y < x + 0
y> -x + 3

75. y ≥- 2
x > 1
24. y ≥ -2
y ≥ 2x + 3

76. y – 2x > 1
y > 1 + 2x
y – 2x < 3
y < 3 + 2x
8x + 5y ≤ 40
5y ≤ 40 – 8x
y ≤ 8 – 8/5x
x + 2y ≤ 8
y ≤ 4 – 1/2x
x≥0
y≥0

77. REVIEW
PAGE(S): 200
NUMBERS: all

78. Activation
Review yesterday’s warm-up

79. 4-8 Using Linear Programming
EQ: What is linear programming?

80. Activation x – 2y < 6 y ≤ -3/2 x + 5 x > 0 y > 0

81. VOCABULARY:
Linear programming– identifies minimum or maximum of a given situation
Constraints—the linear inequalities that are determined by the problem
Objective—the equation that proves the minimum or maximum value.

82. Directions: Read the problem List the constraints List the objective
Graph the inequalities finding the feasible region
Solve for the vertices (the points of intersection)
Test the vertices in the objective

83. Example:
What values of y maximize P given Constraints: y≥3/2x -3 y ≤-x + 7 x≥0 y≥0 Objective: P = 3x +2y x y P

84. You are selling cases of mixed nuts and roasted peanuts
You are selling cases of mixed nuts and roasted peanuts. You can order no more than a total of 500 cans and packages and spend no more than $600. If both sell equally well, how can you maximize the profit assuming you will sell everything that you buy?
items
cost
Profit per case
Mixed Nuts (x items per case) 12 24
Peanuts (y items per case) 15
Totals
500
600
items
cost
Profit per case
Mixed Nuts (x items per case) 12 24 18
Peanuts (y items per case) 15
Totals
500
600 P
items
cost
Profit per case
Mixed Nuts (x items per case) 12
Peanuts (y items per case) 15
Totals
500
items
cost
Profit per case
Mixed Nuts (x items per case)
Peanuts (y items per case)
Totals
Constraints:
x≥0
y≥0
12x + 20y ≤ 500
y ≤ 25 – 3/5 x
24x + 15y ≤ 600
y ≤ 40 – 8/5 x
Objective:
P = 18x + 15y x y P

85. Partner Problem (sample was #8)
A florist has to order roses and carnations for Valentine’s Day. The florist needs to decide how many dozen roses and carnations should be ordered to obtain a maximum profit. Roses: The florist’s cost is $20 per dozen, the profit over cost is $20 per dozen. Carnations: The florist’s cost is $5 per dozen, the profit over cost is $8 per dozen. The florist can order no more than 60 dozen flowers. Based on previous years, a minimum of 20 dozen carnations must be ordered. The florist cannot order more than $450 worth of roses and carnations. Find out how many dozen of each the florist should order to max. profit!
Cost
Total ordered
Profit
Roses (x) 20 1
Carnations(y) 5 8
Totals
450 60 ?
Constraints:
x≥0
y≥20
20 x + 5y ≤450
y≤ 90-5x
x + y ≤60
y≤ 60-x x y
P=20x + 8y 20
20(0)+8(20)=160 60
20(0)+8(60)=480 10 50
20(10)+8(50)=600
17.5=17
(20)17+8(20)=500
Objective:
P= 20x + 8y

86. Sample of what must be handed in for Partner problem

87. A furniture company makes wooden desks and chairs
A furniture company makes wooden desks and chairs. Carpenters and finishers work on each item. On average, the carpenters spend four hours working on each chair and eight hours on each desk. There are enough carpenters for up to 8000 man-hours per week. The finishers spend about two hours on each chair and one hour on each desk. There are enough finishers for a maximum of 1300 man-hours per week. Each chair yields a profit of $150 and each desk $200. Find the maximum amount of profit possible.
carpent
finisher
Profit
chairs(x) 4 2
150
desks(y) 8 1
200
Totals
8000
1300
Constraints:
x≥0
y≥0
4 x + 8y ≤8000
y≤ /2 x
2x + y ≤1300
y≤ x
Objective:
P= 150x + 200y

88. 4-8 Partner PROJECT
See worksheet