Transcript of Systems. Day 1 Systems of Linear Equations System of Linear Equations: two or more linear equations...
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- Systems
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- Day 1
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- Systems of Linear Equations System of Linear Equations: two or
more linear equations together The solution of the system of
equations the point of intersection of the two lines (makes both
equations true) =
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- Example: Solve by Graphing a.) Step 1: Graph both equations on
the same coordinate plane Step 2: Find point of intersection (2,1)
Step 3: Check Answer The solution is (2,1)
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- Example: Solve by graphing b.) Step 1: Graph both equations on
the same coordinate plane Step 2: Find point of intersection (-1,4)
Step 3: Check Answer The solution is (-1,4)
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- Infinite Solutions & No Solutions A system of linear
equations has NO SOLUTION when the graphs of the equations are
parallel (same slope & different y-intercept). A system of
linear equations has an INFINITE NUMBER OF SOLUTIONS when the
graphs of the equations are the same line (same slope &
y-intercept).
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- Example: Solve by graphing a.) have to put in slope-intercept
form Equations have same slopes & different y-intercepts;
therefore they are parallel Answer: No Solution
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- Example: Solve by graphing b.) have to put in slope-intercept
form Equations have the same slope & same y-intercept;
therefore, they are the same line Answer: Infinite # of
Solutions
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- Day 2
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- Example: Solve by Graphing a.) Step 1: Graph both equations on
the same coordinate plane Step 2: Find point of intersection (1,5)
Step 3: Check Answer The solution is (1,5)
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- Example: Solve by Graphing b.)have to put in slope-intercept
form Equations have same slopes & different y-intercepts;
therefore they are parallel Answer: No Solution
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- Example: Solve by Graphing c.)have to put in slope-intercept
form Equations have same slopes & same y-intercepts; therefore
they are the same line Answer: Infinite # Solutions
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- Day 1
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- Example: Solve using Substitution a.) Step 1: Get one of the
variables by itself on one side of the equation Step 2: Plug into
the OTHER equation & solve for variable Step 3: Plug answer
back into EITHER of the original equations to get 2 nd variable
Solution:
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- Example: b.) Step 1: Get one of the variables by itself on one
side of the equation Step 2: Plug into the OTHER equation &
solve for variable Step 3: Plug answer back into EITHER of the
original equations to get 2 nd variable Solution:
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- Example: c.) Need to get a variable by itself Now, our
equations are: or Solution:
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- Example: d.) Need to get a variable by itself (doesnt always
have to be y) Now, our equations are: Solution:
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- Homework Worksheet: Solve by Substitution #1-8
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- Day 2
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- Example: Solve using substitution a.) Variables cancelled out.
Left with a true statement? TRUE! Answer: Variables cancelled out.
Left with a true statement? NOT TRUE! Answer: NO Solution b.)
Infinite # of Solutions
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- Homework Worksheet: Solve by Substitution #9-21
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- Example: Your school committee is planning an field trip for
193 students. There are eight drivers available and two types of
vehicles, school buses and minivans. The school buses seat 51
people each and the minivans seat 8 people each. How many buses and
minivans will be needed?
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- Example: You have 11 bills in your wallet, some are $5 bills
and some are $10 bills. You have a total of $95 in your wallet. How
many $5 bills and how many $10 bills do you have?
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- Day 1
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- Example: Solve using Elimination a.) Step 1: Get one pair of
variables that will cancel -6y and 6y will cancel Step 2: Add
equations & solve for remaining variable Step 3: Plug 1 st
variable into EITHER equation to get 2 nd variable Solution:
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- Example: b.) Step 1: Multiply one equation by a # to get a pair
of variables that will cancel Step 2: Add equations & solve for
remaining variable Step 3: Plug 1 st variable into EITHER equation
to get 2 nd variable Solution:
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- Example: c.) Have to multiply BOTH equations to get a variable
to cancel Can plug back into ANY of the equations Solution:
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- Homework: Worksheet: Solve by Elimination #1-8
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- Day 2
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- Example: Solving using elimination a.)b.)
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- Homework Worksheet: Solving by Elimination #9-21
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- Solving by Substitution
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- Systems with Three Variables The graph of any equation in the
form Ax + By + Cz = D is a plane. The solution of a three-variable
system is the intersection of the three planes.
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- When the solution of a system of equations in 3 variables is
represented by one point, you can write it as an ordered triple:
(x, y, z) (alphabetical order)
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- Example: Solve using substitution a.) Step 1: Choose one
equation to solve for one of its variables Step 2: Substitute the
expression into each of the other equations. Step 3: Write the two
new equations as a system. Solve for both variables. Step 4: Plug
those variables to one of the original equations to get remaining
variable.
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- Example: Solve using substitution b.) Step 1: Choose one
equation to solve for one of its variables Step 2: Substitute the
expression into each of the other equations. Step 3: Write the two
new equations as a system. Solve for both variables. Step 4: Plug
those variables to one of the original equations to get remaining
variable.
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- Solving by Elimination
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- Example: Solve using elimination a.) Step 1: Pair the equations
to eliminate y, since the y- terms are already additive inverses.
Add the equations. Step 2: Write the two new equations as a system,
solve for the other two variables. Step 3: Substitute values in one
of the original equations to solve for last remaining
variable.
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- Example: Solve using elimination b.) Step 2: Pair the equations
to eliminate y, since the y- terms are already additive inverses.
Add the equations. Step 3: Write the two new equations as a system,
solve for the other two variables. Step 4: Substitute values in one
of the original equations to solve for last remaining variable.
Step 1: Find the LCM for the coefficients of the variable you want
to cancel & multiply the equations.
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- System of Linear Inequalities Shading: Plug in (0,0) NOT True!
Shade on side without (0,0) NOT True! Shade on side without (0,0)
The solution is where the shading overlaps Solution
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- Example: Solve by Graphing a.) use slope-int. form use x &
y intercepts x=4y=3 Shading: Plug in (0,0) True! Solution!!!
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- Example: b.) x=2y=1 Shading: Plug in (0,0) NOT True! True!
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- Homework Worksheet: Solving Systems of Inequalities w/ 2
Equations
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- Word Problems
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- Example: A zoo keeper wants to fence a rectangular habitat for
goats. The length should be at least 80ft & the distance around
it should be no more than 310 ft. What are possible dimensions? x =
width of habitat y = length of habitat x=155 y=155 20 100 60 140 20
60 100 140 Shading: Plug in (0,0) NOT True! True! Length Width
Possible dimensions: Means: Width is 20 & Length is 100
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- Example: Suppose you want to fence in a rectangular garden. The
length needs to be at least 50 ft & the perimeter to be no more
than 140 ft. Solve by graphing. x = width of garden y = length of
garden 10 50 30 70 10 30 50 70 x=70 y=70 Shading: Plug in (0,0)
Length Width NOT True! True!
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- Example: Solve by Graphing a.)
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- Example: Solve by Graphing b.)
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- Recall: +h moves LEFT -h moves RIGHT +k moves UP -k moves DOWN
+ V faces up - V faces down
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- Example: Solve by Graphing c.)
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- Example: Solve by Graphing d.)
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- Day 1
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- Linear Programming: a technique that identifies the minimum or
maximum value of some quantity. This quantity is modeled with an
objective function. Limits on the variables in the objective
function are constraints, written as linear inequalities.
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- Example: Suppose you want to buy some tapes & CDs. You can
afford as many as 10 tapes and 7 CDs. You want at least 4 CDs &
at least 10 hours of recorded music. Each tape holds about 45
minutes of music and each CD holds about an hour. a.) Write a
system of inequalities. x = #tapes purchased y = #CDs purchased as
many as 10 tapes as many as 7 Cds at least 4 CDs at least 10 hours
These inequalities model the constraints on x & y.
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- b.) Graph the system of inequalities
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- The shaded region in the graph is the feasible region & it
contains all the points that satisfy all the constraints.
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- Say you buy tapes at $8 each & CDs at $12 each. The
objective function for total cost C is If your total cost is $140,
the equation would be 140 = 8x +12y, shown by the yellow line If
your total cost is $112, the equation would be 112 = 8x +12y, shown
by the purple line
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- As you can see, graphs of the objective function for various
values of C are parallel. Lines closer to the origin (0, 0)
represent lower costs. The graph closest to the origin that
intersects the feasible region intersects it at the vertex (8, 4).
The graph farthest from the origin that intersects the feasible
region intersects it at the vertex (10, 7).
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- Graphs of an objective function that represent a maximum or
minimum value intersect a feasible region at a vertex. Vertex
Principle of Linear Programming If there is a maximum or a minimum
value of the linear objective function, it occurs at one or more
vertices of the feasible region.
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- Example: Find the values of x & y that maximize and
minimize P for the objective function What is the value of P at
each vertex? Constraints Step 1: Graph the Constraints
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- Example: Find the values of x & y that maximize and
minimize P for the objective function What is the value of P at
each vertex? Step 2: Find coordinates for each vertex Vertex
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- Example: Find the values of x & y that maximize and
minimize P for the objective function What is the value of P at
each vertex? Step 3: Evaluate P at each vertex Vertex When x = 4
and y = 3, P has its maximum value of 18. When x = 0 and y = 0, P
has its minimum value of 0.
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- Homework: Worksheet: Textbook page 138 #1-3 Graph is already
done for you, just have to do steps #2 & 3 (all work on
separate paper)
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- Word Problem
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- Example: Suppose you are selling cases of mixed nuts and
roasted peanuts. You can order no more than 500 cans and packages
& spend no more than $600. How can you maximize your profit?
How much is the maximum profit?
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- Continued. Define variables Write Constraints x = # of cases of
mix nuts ordered y = # of cases of roasted peanuts ordered no more
than a total of 500 cans/packages spend no more than $600 can we
have negatives?
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- Continued Write objective function We need to write an equation
for the profit, since that is what we are trying to maximize.
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- Continued Graph constraints reduce!10203040 10 20 30 40
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- Continued Find & test vertices 10203040 10 20 30 40 You can
maximize the profit by selling 15 cases of mixed nuts & 16
cases of roasted peanuts. The maximum profit is $510.
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- Homework Worksheet 3-4 Use graph paper!