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
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System of linear equations
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One solution
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No solutions
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System is consistent … equations are independent
System is inconsistent … equations still independent
Many (infinite) solutions
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System is consistent … equations are dependent
"Elimination" Solution Method
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Given system
2 x  y  15
x y 0
2 x  y  15
x y0
3 x  0 y  15
x5
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
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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
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Linear inequality in two variables written as
ax+by≤c
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Note ≤ could also be <, >, or ≥
Graph of a linear inequality is a "half plane"
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Represents all ordered pairs which satisfy the
inequality
Example
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Given 2x + 3y ≤ 6
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• 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
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Determine if that ordered pair satisfies the inequality
If so – that's the side, if not – other side
Systems of Inequalities
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We seek the ordered pairs which satisfy all
inequalities
Try this system
x y 3
x y 3
Application
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A rectangular pen for
Snidly's pet monster is to
be made out of 40 ft of fence
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Let y = length, x = width
We know 2 x  2 y  40 and
Which sides of the
lines are included?
What is this point?
yx
Application
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What dimensions give an area of 91 ft2 ?
2 x  2 y  40
x  y  91
y
91
x
Application
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What is the formula for
A in terms of y?
2 x  2 y  40
y  20  x
A  y  x   20  x   x
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Graph A
What is the maximum area possible for the
pen?
Assignment A
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Lesson 6.2A
Page 477
Exercises 1 – 67 odd
Linear Programming
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Procedure used to optimize quantities such as
cost and profit
Consists of
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Linear objective function
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Describes a quantity to be optimized
System of linear inequalities called constraints
Solution is set of feasible solutions
Linear Programming Example
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Company produces 2 products
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Constraints
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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
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$35 per CD player
$15 per radio
We need a linear objective
function – what is a function
which gives profit?
Linear Programming Example
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Let radios be x, CD players be y
Profit = 15x + 35y
Constraints
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x≥5
x ≤ 25
x≤y
y ≤ 30
Now determine
vertices of region
(5,30)
(25,30)
(25,25)
(5,5)
Linear Programming Example
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Next plug those vertex ordered pairs into the
profit function
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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
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If the optimal value exists
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It will occur at a vertex of the region of feasible
solutions
Try It Out
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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
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We are buying filing cabinets.
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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
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We seek maximum storage capacity
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What are constraints?
What is the linear objective function?
Graph and solve?
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Assignment B
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Lesson 6.2B
Page 480
Exercises 75 – 91 odd