Linear-Combination Situations

1. SECTION 3-3
Linear-Combination Situations

2. WARM-UP
At a bake sale, bread costs $2 a loaf and cakes coast $3 each.
Suppose you want to buy some bread and cake and spend
exactly $20. How many different combinations of bread and
cake can you buy?

3. WARM-UP
At a bake sale, bread costs $2 a loaf and cakes coast $3 each.
Suppose you want to buy some bread and cake and spend
exactly $20. How many different combinations of bread and
cake can you buy?
B = Bread, C = Cake, A = Total Amount

4. WARM-UP
At a bake sale, bread costs $2 a loaf and cakes coast $3 each.
Suppose you want to buy some bread and cake and spend
exactly $20. How many different combinations of bread and
cake can you buy?
B = Bread, C = Cake, A = Total Amount
2B + 3C = 20

5. WARM-UP
At a bake sale, bread costs $2 a loaf and cakes coast $3 each.
Suppose you want to buy some bread and cake and spend
exactly $20. How many different combinations of bread and
cake can you buy?
B = Bread, C = Cake, A = Total Amount
2B + 3C = 20
3 Combinations

6. WARM-UP
At a bake sale, bread costs $2 a loaf and cakes coast $3 each.
Suppose you want to buy some bread and cake and spend
exactly $20. How many different combinations of bread and
cake can you buy?
B = Bread, C = Cake, A = Total Amount
2B + 3C = 20
3 Combinations
7 loaves of bread, 2 cakes

7. WARM-UP
At a bake sale, bread costs $2 a loaf and cakes coast $3 each.
Suppose you want to buy some bread and cake and spend
exactly $20. How many different combinations of bread and
cake can you buy?
B = Bread, C = Cake, A = Total Amount
2B + 3C = 20
3 Combinations
7 loaves of bread, 2 cakes
4 loaves of bread, 4 cakes

8. WARM-UP
At a bake sale, bread costs $2 a loaf and cakes coast $3 each.
Suppose you want to buy some bread and cake and spend
exactly $20. How many different combinations of bread and
cake can you buy?
B = Bread, C = Cake, A = Total Amount
2B + 3C = 20
3 Combinations
7 loaves of bread, 2 cakes
4 loaves of bread, 4 cakes
1 loaves of bread, 6 cakes

9. LINEAR COMBINATION

10. LINEAR COMBINATION
An expression of the form Ax + By, where A and B
are constants

11. LINEAR COMBINATION
An expression of the form Ax + By, where A and B
are constants
All variables are to the first degree

12. LINEAR COMBINATION
An expression of the form Ax + By, where A and B
are constants
All variables are to the first degree
The variables are not multiplying or dividing each
other

13. LINEAR COMBINATION
An expression of the form Ax + By, where A and B
are constants
All variables are to the first degree
The variables are not multiplying or dividing each
other
Gives the graph of a line (or line segment)

14. EXAMPLE 1
Matt Mitarnowski sold $36 worth of tickets for the Big Guy
Club Picnic. Adult tickets cost $5 and kid tickets cost $2.
a. Write an equation

15. EXAMPLE 1
Matt Mitarnowski sold $36 worth of tickets for the Big Guy
Club Picnic. Adult tickets cost $5 and kid tickets cost $2.
a. Write an equation
A = Adult tickets, K = Kid tickets, T = Total amount

16. EXAMPLE 1
Matt Mitarnowski sold $36 worth of tickets for the Big Guy
Club Picnic. Adult tickets cost $5 and kid tickets cost $2.
a. Write an equation
A = Adult tickets, K = Kid tickets, T = Total amount
T = 5A + 2K

17. EXAMPLE 1
Matt Mitarnowski sold $36 worth of tickets for the Big Guy
Club Picnic. Adult tickets cost $5 and kid tickets cost $2.
a. Write an equation
A = Adult tickets, K = Kid tickets, T = Total amount
T = 5A + 2K
36 = 5A + 2K

18. EXAMPLE 1
Matt Mitarnowski sold $36 worth of tickets for the Big Guy
Club Picnic. Adult tickets cost $5 and kid tickets cost $2.
a. Write an equation
A = Adult tickets, K = Kid tickets, T = Total amount
T = 5A + 2K
36 = 5A + 2K
b. Make a table and graph.

19. EXAMPLE 1
Matt Mitarnowski sold $36 worth of tickets for the Big Guy
Club Picnic. Adult tickets cost $5 and kid tickets cost $2.
a. Write an equation
A = Adult tickets, K = Kid tickets, T = Total amount
T = 5A + 2K
36 = 5A + 2K
b. Make a table and graph.
K 18 13 8 3
A 0 2 4 6

20. EXAMPLE 1
8
6
Kid Tickets
4
2
0
0 4.5 9.0 13.5 18.0
Adult Tickets

21. EXAMPLE 2
Mixtures A and B contain weed killer and water. Mixture A is
5% weed killer and mixture B is 15% weed killer.
a. Write an equation

22. EXAMPLE 2
Mixtures A and B contain weed killer and water. Mixture A is
5% weed killer and mixture B is 15% weed killer.
a. Write an equation
A = Mixture A, B = Mixture B, T = Total

23. EXAMPLE 2
Mixtures A and B contain weed killer and water. Mixture A is
5% weed killer and mixture B is 15% weed killer.
a. Write an equation
A = Mixture A, B = Mixture B, T = Total
.05A + .15B = T

24. EXAMPLE 2
Mixtures A and B contain weed killer and water. Mixture A is
5% weed killer and mixture B is 15% weed killer.
a. Write an equation
A = Mixture A, B = Mixture B, T = Total
.05A + .15B = T
b. How many ounces of 5% solution must be added to 50 oz.
of 15% solution to get 12 oz. of weed killer?

25. EXAMPLE 2
Mixtures A and B contain weed killer and water. Mixture A is
5% weed killer and mixture B is 15% weed killer.
a. Write an equation
A = Mixture A, B = Mixture B, T = Total
.05A + .15B = T
b. How many ounces of 5% solution must be added to 50 oz.
of 15% solution to get 12 oz. of weed killer?
.05A + .15(50) = 12

26. EXAMPLE 2
Mixtures A and B contain weed killer and water. Mixture A is
5% weed killer and mixture B is 15% weed killer.
a. Write an equation
A = Mixture A, B = Mixture B, T = Total
.05A + .15B = T
b. How many ounces of 5% solution must be added to 50 oz.
of 15% solution to get 12 oz. of weed killer?
.05A + .15(50) = 12
.05A + 7.5 = 12

27. EXAMPLE 2
Mixtures A and B contain weed killer and water. Mixture A is
5% weed killer and mixture B is 15% weed killer.
a. Write an equation
A = Mixture A, B = Mixture B, T = Total
.05A + .15B = T
b. How many ounces of 5% solution must be added to 50 oz.
of 15% solution to get 12 oz. of weed killer?
.05A + .15(50) = 12
.05A + 7.5 = 12
.05A = 4.5

28. EXAMPLE 2
Mixtures A and B contain weed killer and water. Mixture A is
5% weed killer and mixture B is 15% weed killer.
a. Write an equation
A = Mixture A, B = Mixture B, T = Total
.05A + .15B = T
b. How many ounces of 5% solution must be added to 50 oz.
of 15% solution to get 12 oz. of weed killer?
.05A + .15(50) = 12
.05A + 7.5 = 12
.05A = 4.5
A = 90 oz

29. EXAMPLE 2
Mixtures A and B contain weed killer and water. Mixture A is
5% weed killer and mixture B is 15% weed killer.
a. Write an equation
A = Mixture A, B = Mixture B, T = Total
.05A + .15B = T
b. How many ounces of 5% solution must be added to 50 oz.
of 15% solution to get 12 oz. of weed killer?
.05A + .15(50) = 12
.05A + 7.5 = 12
.05A = 4.5
A = 90 oz of Mixture A must be added

30. EXAMPLE 2
c. Graph a situation where the final mixture gives 12 ounces
of weed killer in the final mix.

31. EXAMPLE 2
c. Graph a situation where the final mixture gives 12 ounces
of weed killer in the final mix.
A B
0 80
26 0

32. EXAMPLE 2
c. Graph a situation where the final mixture gives 12 ounces
of weed killer in the final mix.
A B
80
0 80
60
26 0
Mixture B
40
20
0
0 75 150 225 300
Mixture A

33. EXAMPLE 2
c. Graph a situation where the final mixture gives 12 ounces
of weed killer in the final mix.
A B
80
0 80
60
26 0
Mixture B
40
20
0
0 75 150 225 300
Mixture A

34. HOMEWORK

35. HOMEWORK
p. 155 #1 - 19
“Hold yourself responsible for a higher standard than anyone
else expects of you. Never excuse yourself.”
-Henry Ward Beecher