⎩ ⎨ ⎧ ≤ - ≤ ≤ - ≤ 42 1 42 1 y x ⎩ ⎨ ⎧

Algebra II Pre-AP -- Assignment 13
Linear Programming -- Graphing Applications of Linear Systems
Class Example: A nutritionist is requested to devise a formula for a base for an instant breakfast product. The
breakfast must contain at least 12 g of protein and 8 g of carbohydrates. A tablespoon of protein powder made
from soybeans has 5 g protein and 2 g carbohydrates. A tablespoon of protein powder made from milk solids
has 2 g protein and 4 g carbohydrates. Soybean protein powder costs $0.70 per tablespoon, and milk protein
powder costs $0.30 per tablespoon. Determine the number of tablespoons of each type of protein powder that
should be used to meet the given requirements and minimize the cost.
Now use linear programming to solve #1 - 4 below. Be sure to use the variables as defined here. You may
use your own graph paper or print the customized grids from Mr. Russell's web site.
1) Kay grows and sells tomatoes (x) and green beans (y). It costs $1 to grow a bushel of tomatoes, and it
takes 1 square yard of land. It costs $3 to grow a bushel of beans, and it takes 6 square yards of land. Kay's
budget is $15, and she has 24 square yards of land available. If she makes $1 profit on each bushel of
tomatoes and $4 profit on each bushel of beans, then how many bushels of each should she grow in order to
maximize profit?
2) A biologist needs at least 40 fish for her experiment. She cannot use more than 25 perch (x) or more than
30 bass (y). Each perch costs $5 and each bass costs $3. How many of each type of fish should she use in
order to minimize her costs?
3) A tray of corn muffins (x) requires 4 cups of milk and 3 cups of flour. A tray of bran muffins (y) takes 2 cups
of milk and 3 cups of flour. There are 16 cups of milk and 15 cups of flour available, and the baker makes a
profit of $3 per tray of corn muffins and $2 per tray of bran muffins. How many trays of each should he make in
order to maximize profits?
4) A dressmaking shop makes dresses (x) and pantsuits (y). The equipment in the shop allows for making at
most 30 dresses and 20 pantsuits in a week. It takes 10 worker-hours to make a dress and 20 worker-hours to
make a pantsuit. There are 500 worker-hours available per week. How many of each should be made if:
a) The profit on a dress and the profit on a pantsuit are the same?
b) The profit on a pantsuit is three times the profit on a dress?
1  x  2  4
5) Graph the system: 
1  y  2  4
(Hint: You
 5 x  2  xy
3x  5 xy  46
6) Solve: 
may want to used what you learned about absolute
value and distance.)
Algebra II Pre-AP -- Assignment 14
Systems of 3 Equations in 3 Variables
Solve #1 - 6 using substitution or elimination. One is inconsistent (no solutions) and one is dependent
(infinitely many solutions.)
 x  y  3z  0

1)  4 x  y  4
 x  6z  2

 3 x  y  z  14

2)  x  2 y  3z  9
5 x  y  5z  30

 3x  y  2 z  10

3) 6 x  2 y  z  2
 x  4 y  3z  7

 x  5 y  2 z  1
 3x  y  2 z  4
 3x  z  13



4)  x  2 y  z  6
5)  2 x  y  3z  10
6) 2 y  3z  12
 2 x  7 y  3 z  7
6 x  2 y  4 z  8
 3x  y  9



8) If y  3 x and z  7 y , then find x  y  z in terms
3x  2 y  z  19
7) In the system 
, what is the
of x.
3 x  y  2 z  13
value of 2 z  2 y ?

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