1. I CAN . . .
~solve systems of equations by
graphing.
lesson 5.1

2. vocabulary { New Word Visual Clue Definition system of equations
system of equations
solution of a system
coincident
more than one equation
x + y = 9
x – y = -5
More than one equation
Point that is true for all equations in the system
Two lines that have the same slope and same y-intercept

3. Graphing systems
Step 1: Equations should be in slope-intercept form. ______________
Rewrite equation if needed.
Step 2: ________ both lines.
Step 3: Find the coordinates of the point where the two lines
____________. (touch)
Graph
intersect

4. Example #1 (0, 3) 3 Solution: _________ or x = ___, y = ___
Find the coordinates of the point where the lines touch.
(0, 3) 3
Solution: _________ or x = ___, y = ___

5. Example #2 2 –1 (2, –1) Solution: ________ or x = ___, y = ____
Find the coordinates of the point where the lines touch.
(2, –1) 2 –1
Solution: ________ or x = ___, y = ____

6. Example #3 Parallel Solution: _______________ No solution
Find the coordinates of the point where the lines touch.
Solution: _______________
No solution

7. Example #4 They overlap! 2y = –4x +14 Solution: _________________
–4x –4x
2y = –4x +14
They overlap!
The lines are coincident (identical)
Solution: _________________
Infinitely many

8. Your turn! One a) b) Parallel lines = No solution c)
Same line = Coincident = Infinitely many
Example 1-1b

9. Draw a picture for each type of solution.
Are you a master?
Draw a picture for each type of solution.
No solution One Solution Infinitely Many

10. vocabulary { New Word Visual Clue Definition solve
solve
3x + 2 = 14
Solve for x
x = 4
To find a value for each variable

11. I CAN . . .
~solve systems of equations by
substitution.
lesson 5.2

12. substitution
Step 1: ___________ to get _______ equation with _____ variable.
Step 2: _______ the equation for the first variable.
Step 3: ____________ to find the second variable.
Substitute
one
one
Solve
Substitute

13. Example #1 5 5 2x + y = 15 2x + y = 15 (y) 2x + (3x) = 15 5x = 15
1. Substitute to get one equation
with one variable
2x + (3x) = 15
5x = 15
2. Solve the equation for the 1st variable
x = 3
3. Substitute to find the
second variable
2x + y = 15
2(3) + y = 15

14. Example #1 2(3) + y = 15 6 + y = 15 –6 –6 y = 9
3. Substitute to find the
second variable
2(3) + y = 15
6 + y = 15
– –6
y = 9
Check your answer! Does 9 = 3(3)?
Yes!

15. Example #2 2x – 2 = x + 2 +2 +2 2x = x + 4 –x –x x = 4
1. One equation with
one variable
2x – 2 = x + 2
2. Solve
2x = x + 4
–x –x
x = 4

16. Example #2 Given: y = x + 2 x y = 4 + 2 y = 6 x = 4
3. Substitute & solve
for the other
x = 4
y = x + 2 x
y = 4 + 2
y = 6
Check your answer! Does 6 = 4 + 2?
Yes!

17. Example #3 Solution: (–1, 5) 5x + 2(3x + 8) = 5 y = 3x + 8
–16 –16
11x = –11
y = 5
x = –1
Solution: (–1, 5)

18. Example #4
y = –2x + 7
Solution: (0, 7)

19. Example #5
Solution:

20. Example #6
Monday’s homework and Tuesday’s homework have a total of 24 problems. There are three times as many problems Tuesday as on Monday. How many are in each set?
Write two equations that fit this situation.
M + T = 24
T = 3M
Solution: Monday had 6 problems & Tuesday had 18.

21. Are you a master?
What do you look for in a system of equations that would lend itself to using substitution instead of graphing?

22. Warm-up
Use substitution to solve
3x + y = 6
-2x + 6y = 4

23. lesson 5.3 Learning Target :I CAN . . .
~solve systems of equations with elimination.
lesson 5.3

24. Elimination of a variable
Step 1: Start with x and y on the same side (standard form). _______________ Step 2: Turn it into an __________ problem. Step 3: Add ____________. One variable should cancel out. Step 4: _______ to find remaining variable. Step 5: _____________ to find the other variable.
Ax + By = C
addition
equations
Solve
Substitute

25. Example #1 x + y = 50 x – y = 16 y – y x + y = 50 33 + y = 50 –33 –33
The sum of two numbers is The difference of the numbers is 16. What are the numbers?
x + y = 50
x – y = 16 y
x + y = 50
– y
33 + y = 50
2x = 66
– –33
y = 17
x = 33
Solution: (33, 17)

26. Example #2 –3x –3x + 4y = 12 3x –3x + 4(–15) = 12 –2y = 30 –2 –2
– –2
–3x – 60 = 12
y = –15
–3x = 72
x = –24
Solution: (–24, –15)

27. Example #3 –3y 2x + 3y = 6 3y 2(3) + 3y = 6 6x = 18 6 + 3y = 6 6 6
– –6
x = 3
3y = 0
y = 0
Solution: (3, 0)

28. Example #4
Some friends go to the movies. They purchase2 popcorns and 3 sodas and pay $ The next weekend they go again and purchase 5 popcorns and 3 sodas for $ How much does the popcorn cost?
2p + 3s = 25.50
5p + 3s = 43.50

29. Are you a master?
Determine which method would be best for solving each system (graphing, substitution, elimination) and why? Do not solve.

30. I CAN . . .
~solve systems of equations with multiplication and elimination.
lesson 5.4

31. Warm UP

32. Example #1 2x – y = 6 3x + 4y = –2 8x – 4y = 24 4 11x = 22
3(2) + 4y = –2
x = 2
6 + 4y = –2
– –6
4y = –8
y = –2

33. Example #2 2x + 3y = 6 2 –2x – 4y = –10 x + 2y = 5 –y = –4 2x + 3y = 6
– –1
2x + 3(4) = 6
2x + 12 = 6
y = 4
–12 –12
2x = –6
x = –3

34. Example #3 3 3x + y = 4 – x + 5y = 34 –14y = –98 –14 –14 3x + y = 4
– –14
3x + y = 4
3x + 7 = 4
y = 7
–7 –7
3x = –3
x = –1

35. (Warmup in notes: Example #4 pg 12 2x – 6y = 8 –2x + 6y = 7 0 = 15
x by 2
2x – 6y = 8
–2x + 6y = 7
0 = 15
No solutions
What if we had a true statement?

36. Example #5 2x – 5y = 8 3x + 2y = 31 4x – 10y = 16 2 15x + 10y = 155 5
3(9) + 2y = 31
x = 9
27 + 2y = 31
– –27
2y = 4
y = 2

37. Example #4 pg10
Some friends go to the movies. They purchase2 popcorns and 3 sodas and pay $ The next weekend they go again and purchase 5 popcorns and 3 sodas for $ How much does the popcorn cost?
2p + 3s = 25.50
5p + 3s = 43.50

38. Are you a master?
Complete the multiplication required to set up the following system for elimination. You do NOT need to solve it.

39. Warm-Up 2x + 2y = 8 x = 2 - y c – 4d = 1 2c – 8d = 2
Which solving technique would you choose for the following two problems? Explain IN WRITING (at least 3 sentences) WHY you chose that technique.
2x + 2y = 8
x = 2 - y
c – 4d = 1
2c – 8d = 2

40. lesson 5.5- the weird stuff
I CAN . . .
~solve systems of equations and figure out what the MEANING of their answer is!
lesson 5.5- the weird stuff

41. Example #1 (page 13)
2x + 2y = 8 x = 2 - y

42. Example #2
c – 4d = 1 2c – 8d = 2

43. Example #3
2b – 4c = 8 -3b + 6c = -12

46. Example #3-5

47. Learning Target: Graph inequalities on the coordinate plane
Lesson 5.6
Learning Target:
Graph inequalities on the coordinate plane

48. Steps: 2. If < or > then shade If ≤ or ≥ then shade
1. Graph the given line – draw lightly
2. If < or > then shade
If ≤ or ≥ then shade
3. Shading – make sure you have slope intercept form (y = mx + b)
if ≤, < then shade below
if ≥, > then shade above

49. Example #1
y ≥ 4

50. Example #2

51. Example #3

52. Example #4

53. Are you a master? True or False.
I love solving systems of equations and inequalities. 

54. Tasks for today
Holiday Elf Word Problems