Systems of Equations and Inequalities Lesson 6.2 Recall … Number of Solutions System of linear equations One solution No solutions System is consistent … equations are.
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Transcript Systems of Equations and Inequalities Lesson 6.2 Recall … Number of Solutions System of linear equations One solution No solutions System is consistent … equations are.
Systems of Equations
and Inequalities
Lesson 6.2
Recall … Number of Solutions
System of linear equations
One solution
No solutions
System is consistent … equations are independent
System is inconsistent … equations still independent
Many (infinite) solutions
System is consistent … equations are dependent
"Elimination" Solution Method
Given system
2 x y 15
x y 0
2 x y 15
x y0
3 x 0 y 15
x5
Eliminate one of the variables
by adding the two equations
together
Then solve for remaining
variable
Now substitute result back into one of equations
5 y 0
to determine 2nd variable
5 y
"Elimination" Solution Method
Note results of this method when system is
inconsistent or dependent
Try these …
x 3y 1
4 x 2 y 10
2x 6 y 2
2 x y 10
Hint … multiply both sides
of bottom equation by
some constant
Can you come up with a "rule
of thumb" which tells you when
a system is either inconsistent
or dependent?
Systems of Inequalities
Linear inequality in two variables written as
ax+by≤c
Note ≤ could also be <, >, or ≥
Graph of a linear inequality is a "half plane"
Represents all ordered pairs which satisfy the
inequality
Example
Given 2x + 3y ≤ 6
• Note: ≤ or ≥ means that line of
equation is included – graph as solid.
• Otherwise line is dotted
Solve for y
Graph equation
y ≤ -2/3x +2
Choose ordered
pair from one side
or the other
(0, 0) is an easy choice
Determine if that ordered pair satisfies the inequality
If so – that's the side, if not – other side
Systems of Inequalities
We seek the ordered pairs which satisfy all
inequalities
Try this system
x y 3
x y 3
Application
A rectangular pen for
Snidly's pet monster is to
be made out of 40 ft of fence
Let y = length, x = width
We know 2 x 2 y 40 and
Which sides of the
lines are included?
What is this point?
yx
Application
What dimensions give an area of 91 ft2 ?
2 x 2 y 40
x y 91
y
91
x
Application
What is the formula for
A in terms of y?
2 x 2 y 40
y 20 x
A y x 20 x x
Graph A
What is the maximum area possible for the
pen?
Assignment A
Lesson 6.2A
Page 477
Exercises 1 – 67 odd
Linear Programming
Procedure used to optimize quantities such as
cost and profit
Consists of
Linear objective function
Describes a quantity to be optimized
System of linear inequalities called constraints
Solution is set of feasible solutions
Linear Programming Example
Company produces 2 products
Constraints
CD players
Radios
What linear inequalities
are expressed by these
constraints?
Must produce 5 ≤ radios ≤ 25
Radios produced ≤ CD players produced
CD players produced ≤ 30
Profit
$35 per CD player
$15 per radio
We need a linear objective
function – what is a function
which gives profit?
Linear Programming Example
Let radios be x, CD players be y
Profit = 15x + 35y
Constraints
x≥5
x ≤ 25
x≤y
y ≤ 30
Now determine
vertices of region
(5,30)
(25,30)
(25,25)
(5,5)
Linear Programming Example
Next plug those vertex ordered pairs into the
profit function
Vertex with largest value will be combination to use
Vertex
(5,5)
(25,25)
(25,30)
(5,30)
P = 15x + 35y
250
1250
1425
1125
Fundamental Theorem of
Linear Programming
If the optimal value exists
It will occur at a vertex of the region of feasible
solutions
Try It Out
For the specified function
P = 5x + 3y
Find the maximum and minimum values for the
region given
(2.5, 7)
(6.5, 5)
(3, 2)
(5, 1)
Practice
We are buying filing cabinets.
X costs $100, requires 6 sq ft, holds 8 cu ft
Y costs $200, requires 8 sq ft, holds 12 cu ft
We can spend a max $1400
We only have 72 sq ft of space
We seek maximum storage capacity
What are constraints?
What is the linear objective function?
Graph and solve?
Assignment B
Lesson 6.2B
Page 480
Exercises 75 – 91 odd