1. Chapter 4 Systems of Equations
4.1 Systems of Equations in Two Variables

2. A system of two linear equations in two variables x and y consists of two equations, Ax + By = C and Dx + Ey = F
A solution of a system of linear equations in two variables is an ordered pair (x, y) that satisfies both equations.

3. infinitely many solutions
NUMBER OF SOLUTIONS OF A LINEAR SYSTEM
CONCEPT
SUMMARY y x y x y x
Lines intersect
one solution
Lines are parallel
no solution
Lines coincide
infinitely many solutions

9. Check, by graphing, whether each system of 3 equations has a common solution. If it does, give the solution. If it does not, state that it does not.

10. Check, by graphing, whether each system of 3 equations has a common solution. If it does, give the solution. If it does not, state that it does not.

11. Without graphing, tell whether or not the following system, in which a  b, has a solution, and if so what it is. If it does not have a solution, explain why not.
Explain why your answer above does not depend on the values of a and b, as long as a  b.

12. A: Put each of the equations in the following system into slope intercept form.
B: From your answer to A, and without graphing, tell whether the graphs of the two equations intersect once, do not intersect, or define the same line.
C: Based on your answer to part B, describe a general method for determining the nature of the solution(s) of a linear system without graphing.

13. HW #4.1 Pg Odd, 16-18

14. HW Quiz 4.1 Saturday, April 15, 2017 Pg 161 5 Pg 161 7 Pg 161 16
Pg d

15. 4.2 Solving Systems of Equations

16. Objective: Solve systems of linear equations by substitution

17. Objective: Solve systems of linear equations by substitution

18. A B

19. Objective: Solve systems of linear equations by Linear Combinations

20. Objective: Solve systems of linear equations by Linear Combinations

21. C D

22. 4.3 Using Systems of Equations

23. You and a friend share the driving on a 280 mile trip
You and a friend share the driving on a 280 mile trip. Your average speed is 58 miles per hour. You friend’s average speed is 53 miles per hour. You drive one hour longer than your friend. How many hours did each of you drive?

24. A gardener has two solutions of weedkiller and water
A gardener has two solutions of weedkiller and water. One is 5% weedkiller and the other is 15% weedkiller. The gardener needs 100 L of a solution that is 12% weedkiller. How much of each solution should she use?

25. A freight train leaves Tyler, traveling east at 35 km/h
A freight train leaves Tyler, traveling east at 35 km/h. One hour later a passenger train leaves Tyler, also traveling east on a parallel track at 40 km/h. How far from Tyler will the passenger train catch the freight train?

26. E The band boosters are organizing a trip to a national competition for the 226-member marching band. A bus will hold 70 students and their instruments. A van will hold 8 students and their instruments. A bus costs $280 to rent for the trip. A van costs $70 to rent for the trip. The boosters have $980 to use for transportation. Write a system of equations whose solution is how many buses and vans should be rented. Solve the system.

28. HW #4.2-3 Pg odd, Pg Odd, 34

29. HW Quiz #4.2-3 Saturday, April 15, 2017Pg Pg Pg Pg

30. 4.4 Systems of Equations in Three Variables

31. An Equation with Three Variables
A solution of a system of equations in three variables is called an ordered triple.
What is an ordered triple?
( x, y, z) An example is (2 , -4, 3).
This ordered triple must be true for all the equations in the system.

32. The graph of a linear equation in three variables is a plane
The graph of a linear equation in three variables is a plane. Thus, if a system of equations in three variables has a unique solution, it is a point common to all of the planes.

33. How should we solve a system in three variables??
We will first solve this equation:
x + y + z = 4
x - 2y - z = 1
2x- y -2z = -1
These numbers indicate the equations in the first, second, and third positions. 1 2 3
First, pick a variable to eliminate. We will eliminate x in this system.

34. Solving a system of three equations
Step 1: *(-1) -x- y - z = -4
x- 2y - z = 1
- 3y -2z = -3 1 2
x + y + z = 4
x - 2y - z = 1
2x- y -2z = -1 1 2 3
Step 2: *(-2) -2x- 2y - 2z = -8
2x - y - 2z = -1
-3y - 4z = -9 1 3
Step 3: -3y -2z = -3
-3y -4z = -9

35. Solving a system of three equations continued
Next part of Step 3:
(-3y -2z = -3)*(-1)
-3y -4z = -9
3y + 2z = 3
-3y - 4z =-9
-2z = -6
z = 3
Now just substitute in 3 for z and find the other variables.

36. The Answers You should have gotten: x = 2 y = -1 z = 3
The ordered triple is (2, -1, 3)

37. Let’s do ANOTHER example!
Let’s solve this system of equations:
2x + y + 2z = 11
3x +2y + 2z = 8
x +4y + 3z = 0 1 2 3
Step 1: *(-2) x - 8y - 6z = x + y + 2z =11
-7y - 4z =11 3 1

38. Solving the system of equations continued
Step 2: *(-3) x - 12y - 9z = x + 2y + 2z = y - 7z = 8 3 2
From Step 1
Step 3: ( -7y - 4z = 11) * (-10) (-10y - 7z = 8)* (7) y +40z = y - 49z = z = z = 6

39. Solving the system of equations continued
Now plug in 6 for the variable z to find x and y.
You should get:
x = 2
y = -5
z = 6
The ordered triple is (2, -5, 6)

40. And yet ANOTHER example!
But this one has a special twist!
Systems of three equations can have infinite many solutions or no solutions.
Here is an example with infinite many solutions:
x – 3y + z = x – y – 2z = x + 2y – 3z = -1 1 2 3

41. Solving a system of equations that has infinite many solutions continued
Step 1: *(-3) x - 3y + 9z = x +3y -13z = x - 4z = -20 1 3
Step 2: *(1/2) -8x - 4z = x - 2z = -10
Step 3: -4x - 2z = x + 2z = = 0
Signifies the system of equations has infinite many solutions

42. Summary Remember, solutions are written as an ordered triple.
Remember, solutions can also be no solution or infinite many solutions.
If one equation is missing a variable, just line it up with the other equations.
For example: 3p r = 11
q - 7r = 4
p - 6q = 1
Check your solutions in all equations to make sure it is correct. Just because it answers two of them, doesn’t mean it answers all of them.

44. A B

45. HW #4.4a Pg Odd, 28-30

46. 4.4 Systems of Equations in Three Variables

57. Groups to work on solving

58. HW #4.4b Pg Even

59. HW Quiz 4.4b Saturday, April 15, 2017

60. 4.5 Using a System of Three Equations

61. In yesterday’s swim meet, Roosevelt High dominated in the individual events, with 24 individual-event placers scoring a total of 56 points. A first-place finish scores 5 points, a second-place finish scores 3 points, and a third-place finish scores 1 point. Having as many third-place finishers as first- and second-place finishers combined really shows the team’s depth. Use a system of three equations in 3 variables to determine the number of 1st, 2nd, and 3rd place finishers Roosevelt had.

62. You have $25 to spend on picking 21 pounds of three different types of apples in an orchard. The Empire apples cost $1.40 per pound, the Red Delicious apples cost $1.10 per pound, and the Golden Delicious apples cost $1.30 per pound. You want twice as many Red Delicious apples as the other two kinds combined. Write a system of equations to represent the given information. How many pounds of each type of apple should you buy?

63. Gina sells magazines part time
Gina sells magazines part time. On Thursday, Friday, and Saturday, she sold $66 worth. On Thursday she sold $3 more than on Friday. On Saturday she sold $6 more than on Thursday. How much did she take in each day?

64. Find a three digit positive integer such that the sum of all three digits is 14, the tens digit is two more than the ones digit, and if the digits are reversed the number is unchanged.

65. HW #4.5 Pg Odd, 18-19

66. HW Quiz #4.5 Saturday, April 15, 201711 13 15 17 9

67. 4.6 Consistent and Dependent Systems

68. Objective: Determine whether a system of equations is consistent or inconsistent.
Consistent System: If a system of equations has at least one solution
Inconsistent System: If a system of equations has no solution

69. Objective: Determine whether a system of equations is consistent or inconsistent.

70. Objective: Determine whether a system of equations is dependent

71. Is it possible to have a system that is consistent and dependent?
Is it possible to have a system that is inconsistent and dependent?
Is it possible to have an inconsistent system that is not dependent?
Is it possible to have an consistent system that is not dependent?

72. Objective: Determine whether a system of equations is dependent

73. Determine if the systems are consistent, inconsistent and dependentB A

77. HW #4.6 Pg Odd, 25-31

78. HW Quiz 4.6 Saturday, April 15, 2017

79. 4.7 Systems of Linear Inequalities

80. Objective: Graph a linear inequality
The boundary line of the inequality divides the coordinate plane into two half-planes; a shaded region containing the points that are solutions of the inequality, and an unshaded region which contains the points that are not.

81. Objective: Graph a linear inequality

82. Objective: Graph a system of linear inequalities
A system of linear inequalities is two or more linear inequalities in the same variables and is also called a system of inequalities.
A solution of a system of linear inequalities is an ordered pair that is a solution of each inequality in the system.
The graph of a system of linear inequalities is the graph of all solutions of the system.

83. Objective: Graph a system of linear inequalities

84. Objective: Graph a system of linear inequalities

85. Objective: Graph a system of linear inequalities

86. Objective: Graph a system of linear inequalities

89. HW #4.7 Pg Odd, 40-41

90. 4.8 Using Linear Programming

91. Objective: Solve problems using linear programming.
A bakery is making whole-wheat bread and apple bran muffins. For each batch of bread they make $35 profit. For each batch of muffins they make $10 profit. The bread takes 4 hours to prepare and 1 hour to bake. The muffins take 0.5 hour to prepare and 0.5 hour to bake. The maximum preparation time available is 16 hours. The maximum baking time available is 10 hours. How many batches of bread and muffins should be made to maximize profits?

92. Objective: Solve problems using linear programming.
Optimization means finding the maximum or minimum value of some quantity.
Linear programming is the process of optimizing a linear objective function subject to a system of linear inequalities called constraints.

93. Objective: Solve problems using linear programming.

94. Objective: Solve problems using linear programming.

96. Objective: Solve problems using linear programming.
A bakery is making whole-wheat bread and apple bran muffins. For each batch of bread they make $35 profit. For each batch of muffins they make $10 profit. The bread takes 4 hours to prepare and 1 hour to bake. The muffins take 0.5 hour to prepare and 0.5 hour to bake. The maximum preparation time available is 16 hours. The maximum baking time available is 10 hours. How many batches of bread and muffins should be made to maximize profits?

97. Objective: Solve problems using linear programming.
Bread
(0, 4)
(16, 2)
(0, 0)
(20, 0)
Muffins

98. Objective: Solve problems using linear programming.
Wheels Inc. makes mopeds and bicycles. Experience shows they must produce at least 10 mopeds. The factory can produce at most 60 mopeds per month. The profit on a moped is $134 and on a bicycle, $20. They can make at most 120 units combined. How many of each should they make per month to maximize profit?

99. Objective: Solve problems using linear programming.
(0, 60)
(60, 60)
Mopeds
(0, 10)
(110, 10)
Bikes

100. Objective: Solve problems using linear programming.
Farm Management A farmer has 70 acres of land available for planting either soybeans or wheat. The cost of preparing the soil, the workdays required, and the expected profit per acre planted for each type of crop are given in the following table:
The farmer cannot spend more than $1800 in preparation costs nor use more than a total of 120 workdays. How many acres of each crop should be planted to maximize the profit? What is the maximum profit?

101. (0. 30)
Wheat
(24. 12)
(0. 0)
(30. 0)
Soy

102. A manufacturer of skis produces two types: downhill and cross- country
A manufacturer of skis produces two types: downhill and cross- country. Use the following table to determine how many of each kind of ski should be produced to achieve a maximum profit. What is the maximum profit?

103. HW #4.8 Pg

104. HW Quiz 4.8 Saturday, April 15, 2017

105. Test Review
Part 1

106. Find a and b so that the system below has the unique solution (-2, 3)

107. Find p and q so that the graph of the equation y = x2 + px + q passes through (-1, 3) and (2, 4)

109. Know the terms: Dependent, Consistent, and Inconsistent

110. Part 2
Part two will consist of three linear programming problems along with one proof

111. HW # R-4 Pg 200 1-13 Study All Challenge Problems

112. Find the area of an equilateral triangle