1. HW#1: Systems-Graphing
SWBAT… Solve a systems of equations using the graphing method Wed, 11/30
Agenda
WU: review back of agenda problems (15 min)
Read examples (5 min)
Systems with graphing (20 min)
WARM-UP:
Get ready to learn about systems of equations using the graphing method
HW#1: Systems-Graphing

2. We are starting a new unit: Systems of Linear Equations & Inequalities
SWBAT…
Solve a systems of linear equations using the graphing method
Solve a systems of linear equations using the substitution method
Solve a systems of linear equations using the elimination method
Write systems of equations based on real life scenarios
Solve linear inequalities
Solve systems of linear inequalities

3. For 5 minutes… silently read the first page of the graphing examples

4. Solving Systems of Equations by Graphing
Step 1) Write the equations of the lines in slope-intercept form.
Step 2) Graph each line on the same graph.
Step 3) Determine the point of intersection and write this point as an ordered pair (1 solution.)
If the two equations represent the same line, the system of equations has infinitely many solutions (same line.)
If the two equations have no points in common, the system of equations has no solution (parallel lines.)

5. Example
Graph the system of equations. Determine whether the system has one solution, no solution, or infinitely many solutions. If the system has one solution, determine the solution.
x – y = 2
3y + 2x = 9
Step 1: Write each equation in slope-intercept form.
3y + 2x = 9
- 2x -2x
x – y = 2
+ y +y
3y = -2x + 9
x = 2 + y 3 3 3
x – 2 = y

6. x y
Step 2: Graph each line on the same graph
Step 3: Determine the point of intersection.
y = x – 2
The point of intersection of the two lines is the point (3,1).
This system of equations has one solution, the point (3,1) .

7. Additional Examples
Graph the system of equations. Determine whether the system has one solution, no solution, or infinitely many solutions. If the system has one solution, determine the solution.

8. Conclusions
The solution to a systems of equations is the point where the two lines intersect (one solution)
No solution will be parallel lines
Infinite solution will be the same line

9. HW#2: Systems-Substitution
SWBAT… Solve a systems of equations using the graphing method Thurs, 12/1
Agenda
WU (10 min)
Review HW#1 (25 min)
Quiz: Systems-graphing (10 min)
WARM-UP:
1. Determine whether (1, 2) is a solution of the system:
2x – 3y = -4
2x + y = 4
HW#2: Systems-Substitution
Yes = and 4 = 4

10. Activity-Systems of equations: Graphing
Working in pairs:
Directions: Given your assigned problem, graph the system of equations on a big piece of graph paper. Determine whether the system has one solution, no solution, or infinite solutions. If the system has one solution, name it (see worksheet on how to name it.)
Show your work on your graph paper
After 15 minutes, I will randomly choose 3 groups to present their system

11. Mon, 12/5 SWBAT… solve systems of equations using the graphing method
Agenda
WU (10 min)
Poster presentations (20 min)
Quiz – 3 graphing problems (15 min)
Warm-Up:
Describe the advantages and disadvantages to solving systems of equations by graphing.
Compare m and b
(same or different)
Number of Solutions
One
None
Infinite
Sample Answer: Graphing clearly shows whether a system of equations has one solution, no solution, or infinitely many solutions. However, finding the exact values of x and y from a graph can be difficult. 11

12. HW#1: Systems-Graphing Method Answers:
1 Solution: (1, 2)
1 Solution: (-4, -2)
Infinite Solutions
1 Solution: (-2, -2)
1 Solution: (-3, 5)
Top equation: multiply by 5/8
Bottom equation: multiply by 5/2
1 Solution: (3, -3)
Top equation: multiply by 3/2
Bottom equation: multiply by 5/1

13. 10. Compare m and b Number of Solutions
9. Describe the advantages and disadvantages to solving systems of equations by graphing.
10.
Compare m and b
(same or different)
Number of Solutions
One
None
Infinite
Sample Answer: Graphing clearly shows whether a system of equations has one solution, no solution, or infinitely many solutions. However, finding the exact values of x and y from a graph can be difficult. 13

14. Conclusions
Compare m and b
Number of Solutions
One
None
Infinite

15. (b can be same or different)
Conclusions
Compare m and b
Number of Solutions
Different m values
(b can be same or different)
One
None
Infinite

16. (b can be same or different) Same m value, but different b values
Conclusions
Compare m and b
Number of Solutions
Different m values
(b can be same or different)
One
Same m value, but different b values
None
Infinite

17. Conclusions Compare m and b Number of Solutions Different m values One
(b can be same or different)
One
Same m value, but different b values
None
Same m value and same b value
Infinite

18. Quiz: complete on graph paper
Graph the system of equations. Determine whether the system has one solution, no solution, or infinitely many solutions. If the system has one solution, determine the solution.

19. Problem 1 y The two equations in slope-intercept form are: x
Plot points for each line.
Draw in the lines.
This system of equations represents two intersecting lines.
The solution to this system of equations is a single point (3,0).

20. Problem 2 y The two equations in slope-intercept form are: x
Plot points for each line.
Draw in the lines.
This system of equations represents two parallel lines.
This system of equations has no solution because these two lines have no points in common.

21. Problem 3 y The two equations in slope-intercept form are: x
Plot points for each line.
Draw in the lines.
These two equations represent the same line.
Therefore, this system of equations has infinitely many solutions .

22. 1.) The sum of two numbers is 20.
Tues, 12/6 SWBAT… solve systems of equations using the substitution method
Agenda
WU (10 min)
2 Examples – Systems of equations: substitution method (15 min)
WU: Use substitution to solve the system of equations
1.) The sum of two numbers is 20.
The difference between three times the larger number and twice the smaller is 40.
2.) Verify or check that your solution is correct
HW#2: Substitution method
x + y = 20
3x – 2y = 40
Answer: (4, 16) 22 22

23. Systems of equations: substitution method
The steps using the substitution method are shown in the box on HW#2
Warm-Up: Solve the system using substitution
The sum of two numbers is 20.
The difference between three times the larger number and twice the smaller is 40.
Verify that your solution is correct.
Follow the directions from the box and solve the above system.
x + y = 20
3x – 2y = 40
Answer: (4, 16)

24. 1.) y = 2x – 4 2.) x = y – 1 -6x + 3y = -12 -x + y = -1
Use substitution to solve the system of equations
1.) y = 2x – ) x = y – 1
-6x + 3y = x + y = -1
1.) This statement is an identity. Thus, there are an infinite number of solutions.
If you solve the bottom equation for y, you should notice that the slopes are the same and the y-intercepts are also the same.
2.) No solution
If you solve both equations for y, you should notice that the slopes are the same and they have different y-intercepts 24

25. HW#2: Substitution Answers
(5, 10)
(0, 2)
(2, 0)
No Solution
Infinite Solution
(0, -6)
a.) t = cost of a taco, b = cost of a burrito
b.) 3t + 2b = 7.40
4t + 1b = 6.45
c.) c = 1.1, b = 2.05
d.) The cost of 2 tacos is $2.20 and the cost of 2 burritos is $4.10.

26. HW#3: Elimination method
Wed, 12/7 SWBAT… solve systems of equations using the elimination method
Agenda
WU (10 min)
Review HW#2 (10 min)
Quiz: Systems-substitution method (10 min)
Systems: Elimination method using addition/subtraction (15 min)
WU: Use substitution to solve the system:
Twice one number added to another is 18. Four times the first number minus the other number is 12. Find the numbers.
HW#3: Elimination method
x = first number
y = second number
2x + y = 18
4x – y = 12
x = 5
y = 8 26

27. QUIZ: Solve the system below using the substitution method
x + 4y = 1
2x – 3y = -9
Extra Credit: Find a value of n such that the x-value of the solution of the system below is 4. Show or explain your work.
5x – 10y = 50
nx + 10y = 6
Answer: (-3, 1)
EC: n = 9

28. Ex.1: Elimination using Addition
Negative three times one number plus five times another number is -11.
Three times the first number plus 7 times the other number is -1.
Find the numbers.
-3x + 5y = -11
3x + 7y = -1
Q: What did you notice about the x coefficients?
A: They were the opposite!
(2, -1)

29. Ex. 2: Elimination using Subtraction
Q: What do you notice about the t coefficients?
A: They are the same!
(-4, 7)
-4r=-16
r=4
2t + 5r = 6
2t + 5(4) = 6
2t = -14
t = -7

30. HW#3: Elimination Answers
(5, 2)
(1, 6)
(6, 1)
(-3, 5)
(4, -1)
(2, 3)
(2, -3)

31. HW#1 - HW#4 will be collected and graded tomorrow!
Mon, 12/12 SWBAT… solve a system of equations using elimination using multiplication
Agenda
WU (10 min)
Three examples: elimination using multiplication (20 min)
Practice – HW#4
Warm-Up:
1. Read over the weekly agenda
2. Open your packet to the steps for solving system of equations – elimination using multiplication
HW#1 - HW#4 will be collected and graded tomorrow! 31

32. Ex. 1a: Elimination using Multiplication
(easiest to eliminate the x variables)
5x + 6y = -8
2x + 3y = -5
(2, -3)

33. Ex. 1b: Elimination using Multiplication
(eliminate the y variables)
P.S. You would have to love math to do so!
5x + 6y = -8
2x + 3y = -5
(2, -3) 33

34. Ex 2a: Elimination using Multiplication (Eliminate the x variables)
9x + 5y = 34
8x – 2y = -2
(1, 5)

35. Ex 2b: Elimination using Multiplication (Eliminate the y variables)
9x + 5y = 34
8x – 2y = -2
(1, 5) 35

36. Ex3a: Elimination using Multiplication (Eliminate the x variables)
3x + 3y = 9
4x + 2y = 8
Answer: (1, 2)

37. Ex3b: Elimination using Multiplication (Eliminate the y variables)
3x + 3y = 9
4x + 2y = 8
Answer: (1, 2)

38. HW#4: Elimination Answers
(1, 2)
(6, 18)
c = 3.95, a = 5.95
a.) (4, 1) d.) (0, 3) (2, 5)

39. Tues, 12/13 SWBAT… know when the best time to use each system method
Agenda
WU (5 min)
Concept Summary – Best time to use each method (15 min)
5 examples on which method is best (10 min)
Real life examples (15 min)
Warm Up:
Carefully rip out HW#1 – HW#4
Place in the middle of your tables
Best time to use chart
HW#5 and HW#6 39 39

40. Fill in the chart below:
Method
The Best Time to Use
Graphing
Substitution
Elimination using Addition
Elimination using Subtraction
Elimination using Multiplication

41. Fill in the chart below:
Method
The Best Time to Use
Graphing
To estimate solutions, since graphing usually does not give an exact solution. To visualize the equations.
Substitution
Elimination using Addition
Elimination using Subtraction
Elimination using Multiplication

42. Fill in the chart below:
Method
The Best Time to Use
Graphing
To estimate solutions, since graphing usually does not give an exact solution. To visualize the equations.
Substitution
If one of the variables in either equation has a coefficient of 1.
Elimination using Addition
Elimination using Subtraction
Elimination using Multiplication

43. Fill in the chart below:
Method
The Best Time to Use
Graphing
To estimate solutions, since graphing usually does not give an exact solution. To visualize the equations.
Substitution
If one of the variables in either equation has a coefficient of 1.
Elimination using Addition
If one of the variables has opposite coefficients.
Elimination using Subtraction
Elimination using Multiplication

44. Fill in the chart below:
Method
The Best Time to Use
Graphing
To estimate solutions, since graphing usually does not give an exact solution. To visualize the equations.
Substitution
If one of the variables in either equation has a coefficient of 1.
Elimination using Addition
If one of the variables has opposite coefficients.
Elimination using Subtraction
If one of the variables has the same coefficient.
Elimination using Multiplication

45. Fill in the chart below:
Method
The Best Time to Use
Graphing
To estimate solutions, since graphing usually does not give an exact solution. To visualize the equations.
Substitution
If one of the variables in either equation has a coefficient of 1.
Elimination using Addition
If one of the variables has opposite coefficients.
Elimination using Subtraction
If one of the variables has the same coefficient.
Elimination using Multiplication
If none of the coefficients are 1 and neither of the variables can be eliminated by simply adding or subtracting the equations.

46. Which method is best to use? Why?
1. x = 12y – 14
3y + 2x = -2
Substitution; one equation is solved for x
2. 20x + 3y = 20
-20x + 5y = 60
Elimination using addition to eliminate x
3. y = x + 2
y = -2x + 3
Substitution; both equations are solved for y

47. Which method is best to use? Why?
x + 3y = 20
-20x + 5y = 60
Elimination using subtraction to eliminate x
-5x – 3y = 20
-5x + 3y = 60
Elimination using subtraction to eliminate x OR elimination using addition to eliminate y 47

48. How a customer uses systems of equations to see what he paid
Two groups of students order burritos and tacos at Atontonilco. One order of 3 burritos and 4 tacos costs $ The other order of 9 burritos and 5 tacos costs $ How much did each taco and burrito cost?
B = Burritos
T = Tacos
3B + 4T = 11.33
9B + 5T = 23.56
Each taco costs $1.49 and each burrito costs $1.79.

49. How a fair manager uses systems of equations to plan his inventory
The admission fee at a small fair is $1.50 for children and $4.00 for adults. On a certain day, 2,200 people enter the fair and $5,050 is collected. How many children and how many adults attended?
C = #children
A = #adults
C + A = 2200
1.50C A = 5050
There were 1500 children and 700 adults.

50. How a school uses systems of equations to see how many tickets they sell
Your class sells a total of 64 tickets to the school play. A student ticket costs $1 and an adult ticket costs $ Your class collects $109 in total tickets sales. How many adult and student tickets did you sell?
s = # student tickets
a = # adult tickets
s + a = 64
s + 2.5a = 109
34 student tickets were sold and 30 adult tickets

51. How a customer uses systems of equations to see what he paid
A landscaping company placed two orders with a nursery. The first order was for 13 bushes and 4 trees, and totaled $487. The second order was for 6 bushes and 2 trees, and totaled $232. The bills do not list the per-item price. What were the costs of one bush and of one tree?
B = #bushes
T = #trees
13B + 4T = 487
6B + 2T = 232
Bushes cost $23 each; trees cost $47 each.

52. How a bakery uses systems of equations to track their inventory
La Guadalupana Bakery sells pies for $6.99 and cakes for $ The total number of pies and cakes sold on a busy Friday was 36. If the amount collected for all the pies that day was $331.64, how many of each type were sold?
p = # pies
c = # cakes
p + c = 36
6.99p c =
16 pies were sold and 20 cakes.

53. How a math student uses systems of equations to solve math puzzles!
The sum of two numbers is 25 and their difference is 7. Find the numbers.
x = first number
y = second number
x + y = 25
x – y = 7
x = 16
y = 9

54. How a math student uses systems of equations to solve math puzzles!
Twice one number added to another is 18. Four times the first number minus the other number is 12. Find the numbers.
x = first number
y = second number
2x + y = 18
4x – y = 12
x = 5
y = 8