Vocab
the solution to a system with 2 variables
ordered pair
solve this system using substitution:
x=5
2x + 4y = 22
(5, 3)
solve:
y = x + z + 5
z = -3y - 3
2x - y = -4
(-2, 0, -3)
Which value would be the maximum?
a) 18
b) 24
c)7
b) 24
is the following problem asking to find a maximum or minimum?
A calculator company produces a scientific calculator and a graphing calculator. Long-term projections indicate an expected demand of at least 100 scientific and 80 graphing calculators each day. Because of limitations on production capacity, no more than 200 scientific and 170 graphing calculators can be made daily. To satisfy a shipping contract, a total of at least 200 calculators much be shipped each day. If each scientific calculator sold results in a $2 loss, but each graphing calculator produces a $5 profit, how many of each type should be made daily to maximize net profits?
Maximize
the shaded region in a linear programming problem
feasible region
solve this system using elimination:
x + 3y = 9
2x + 3y = 15
(6, 1)
solve:
-2y + 5z = -3
y = -5x - 4z - 5
x = 4z + 4
(0, -1, -1)
what quadrant do we typically work in when solving linear programming problems?
1st quadrant
define the variables:
A calculator company produces a scientific calculator and a graphing calculator. Long-term projections indicate an expected demand of at least 100 scientific and 80 graphing calculators each day. Because of limitations on production capacity, no more than 200 scientific and 170 graphing calculators can be made daily. To satisfy a shipping contract, a total of at least 200 calculators much be shipped each day.
x = number of scientific calculators produced
y = number of graphing calculators produced
what is the solution to a system with 3 variables called
ordered triple
solve this system using substitution:
-x + 4y = 6
3x - 2y = 2
(2, 2)
solve:
4x + 4y + z = 24
2x - 4y + z = 0
5x - 4y - 5z = 12
(4, 2, 0)
given the objective function and vertices, find the minimum:
f(x,y) = 3x + 2y
(4, 0) (2, - 1) (-5, 9)
(-5, 9)
write the objective function:
If each scientific calculator sold results in a $2 loss, but each graphing calculator produces a $5 profit, how many of each type should be made daily to maximize net profits?
f(x, y) = -2x + 5y
equation that is used to find maximum and minimum values
objective function
solve using elimination:
5x + y = 9
10x - 7y = -18
(1, 4)
solve:
-3x - y - 3z = -8
-5x + 3y + 6z = -4
-6x - 4y + z = -20
(2, 2, 0)
The value of 27 matches with which vertex:
f(x, y) = 5x - y
(4, 7) (6, 3) (10, 2) (5, -3)
(6, 3)
write the constraints:
A calculator company produces a scientific calculator and a graphing calculator. Long-term projections indicate an expected demand of at least 100 scientific and 80 graphing calculators each day. Because of limitations on production capacity, no more than 200 scientific and 170 graphing calculators can be made daily. To satisfy a shipping contract, a total of at least 200 calculators much be shipped each day.
x > 100 (all of these should also be equal to)
y > 80
x < 200
y < 170
x + y > 200
the values the maximize or minimize an objective function
vertices
solve using elimination:
3x - 2y = 2
5x - 5y = 10
(-2, -4)
solve:
-6x - 2y + 2z = -8
3x - 2y - 4z = 8
6x - 2y - 6z = -18
no solution
is (3, 4) a solution for the following feasible region, EXPLAIN:
x > 0
y > 0
2x + 4y < 8
no, (3,4) is not a solution
The objective function is maximized at the vertex (100, 170) when using the objective function f(x, y) = -2x + 5y. Write a conclusion statement: what is the maximum profit and where does it occur?
x = number of scientific calculators sold
y = number of graphing calculators sold.
The company needs to sell 100 scientific calculators and 170 scientific calculators to achieve a maximum profit of $650.