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) =
Slide 4
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)
Slide 5
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)
Slide 6
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).
Slide 7
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
Slide 8
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
Slide 9
Day 2
Slide 10
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)
Slide 11
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
Slide 12
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
Slide 13
Day 1
Slide 14
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:
Slide 15
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:
Slide 16
Example: c.) Need to get a variable by itself Now, our
equations are: or Solution:
Slide 17
Example: d.) Need to get a variable by itself (doesnt always
have to be y) Now, our equations are: Solution:
Slide 18
Homework Worksheet: Solve by Substitution #1-8
Slide 19
Day 2
Slide 20
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
Slide 21
Homework Worksheet: Solve by Substitution #9-21
Slide 22
Slide 23
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?
Slide 24
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?
Slide 25
Day 1
Slide 26
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:
Slide 27
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:
Slide 28
Example: c.) Have to multiply BOTH equations to get a variable
to cancel Can plug back into ANY of the equations Solution:
Slide 29
Homework: Worksheet: Solve by Elimination #1-8
Slide 30
Day 2
Slide 31
Example: Solving using elimination a.)b.)
Slide 32
Homework Worksheet: Solving by Elimination #9-21
Slide 33
Solving by Substitution
Slide 34
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.
Slide 35
Slide 36
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)
Slide 37
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.
Slide 38
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.
Slide 39
Solving by Elimination
Slide 40
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.
Slide 41
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.
Slide 42
Slide 43
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
Slide 44
Example: Solve by Graphing a.) use slope-int. form use x &
y intercepts x=4y=3 Shading: Plug in (0,0) True! Solution!!!
Slide 45
Example: b.) x=2y=1 Shading: Plug in (0,0) NOT True! True!
Slide 46
Homework Worksheet: Solving Systems of Inequalities w/ 2
Equations
Slide 47
Word Problems
Slide 48
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
Slide 49
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!
Slide 50
Slide 51
Example: Solve by Graphing a.)
Slide 52
Example: Solve by Graphing b.)
Slide 53
Recall: +h moves LEFT -h moves RIGHT +k moves UP -k moves DOWN
+ V faces up - V faces down
Slide 54
Example: Solve by Graphing c.)
Slide 55
Example: Solve by Graphing d.)
Slide 56
Day 1
Slide 57
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.
Slide 58
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.
Slide 59
b.) Graph the system of inequalities
Slide 60
The shaded region in the graph is the feasible region & it
contains all the points that satisfy all the constraints.
Slide 61
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
Slide 62
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).
Slide 63
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.
Slide 64
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
Slide 65
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
Slide 66
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.
Slide 67
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)
Slide 68
Word Problem
Slide 69
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?
Slide 70
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?
Slide 71
Continued Write objective function We need to write an equation
for the profit, since that is what we are trying to maximize.
Slide 72
Continued Graph constraints reduce!10203040 10 20 30 40
Slide 73
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.