Chapter 4 Systems of Equations

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Chapter 4Systems of Equations

• 4.1 Systems of Equations in Two Variables

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A system of two linear equations in two variables
x and y consists of two equations, Ax By C
and Dx Ey F
A solution of a system of linear equations in two
variables is an ordered pair (x, y) that
satisfies both equations.
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Lines intersect one solution
Lines are parallel no solution
Lines coincide infinitely many solutions
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Check, by graphing, whether each system of 3
equations has a common solution. If it does, give
the solution. If it does not, state that it does
not.
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Check, by graphing, whether each system of 3
equations has a common solution. If it does, give
the solution. If it does not, state that it does
not.
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Without graphing, tell whether or not the
following system, in which a ? b, has a solution,
and if so what it is. If it does not have a
solution, explain why not.
Explain why your answer above does not depend on
the values of a and b, as long as a ? b.
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A Put each of the equations in the following
system into slope intercept form.
B From your answer to A, and without graphing,
tell whether the graphs of the two equations
intersect once, do not intersect, or define the
same line.
C Based on your answer to part B, describe a
general method for determining the nature of the
solution(s) of a linear system without graphing.
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HW 4.1Pg 161 1-15 Odd, 16-18
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HW Quiz 4.1Wednesday, December 09, 2015
Pg 161 5 Pg 161 7 Pg 161 16 Pg 161 18
Pg 161 9 Pg 161 11 Pg 161 15d Pg 161 18
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4.2 Solving Systems of Equations
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Objective Solve systems of linear equations by
substitution
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Objective Solve systems of linear equations by
substitution
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A
B
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Objective Solve systems of linear equations by
Linear Combinations
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Objective Solve systems of linear equations by
Linear Combinations
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D
C
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4.3 Using Systems of Equations
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You and a friend share the driving on a 280 mile
trip. Your average speed is 58 miles per hour.
You friends average speed is 53 miles per hour.
You drive one hour longer than your friend. How
many hours did each of you drive?
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A gardener has two solutions of weedkiller and
water. One is 5 weedkiller and the other is 15
weedkiller. The gardener needs 100 L of a
solution that is 12 weedkiller. How much of each
solution should she use?
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A freight train leaves Tyler, traveling east at
35 km/h. One hour later a passenger train leaves
Tyler, also traveling east on a parallel track at
40 km/h. How far from Tyler will the passenger
train catch the freight train?
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E The band boosters are organizing a trip to a
national competition for the 226-member marching
band. A bus will hold 70 students and their
instruments. A van will hold 8 students and their
instruments. A bus costs 280 to rent for the
trip. A van costs 70 to rent for the trip. The
boosters have 980 to use for transportation.
Write a system of equations whose solution is how
many buses and vans should be rented. Solve the
system.
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HW 4.2-3Pg 166-167 1-25 odd, 26-29Pg 173 27-33
Odd, 34
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HW Quiz 4.2-3Wednesday, December 09, 2015
Pg 167 28 Pg 167 29 Pg 173 31 Pg 173 34
Pg 167 28 Pg 167 29 Pg 173 34 Pg 173 31
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4.4 Systems of Equations in Three Variables
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An Equation with Three Variables

• A solution of a system of equations in three
  variables is called an ordered triple.
• What is an ordered triple?
• ( x, y, z) An example is (2 , -4, 3).
• This ordered triple must be true for all the
  equations in the system.

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• The graph of a linear equation in three variables
  is a plane. Thus, if a system of equations in
  three variables has a unique solution, it is a
  point common to all of the planes.

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How should we solve a system in three variables??

• We will first solve this equation
• x y z 4
• x - 2y - z 1
• 2x- y -2z -1

These numbers indicate the equations in the
first, second, and third positions.
First, pick a variable to eliminate. We will
eliminate x in this system.
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Solving a system of three equations

• x y z 4
• x - 2y - z 1
• 2x- y -2z -1

Step 3 -3y -2z -3 -3y -4z -9
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Solving a system of three equations continued

• Next part of Step 3

Now just substitute in 3 for z and find the other
variables.
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The Answers

• You should have gotten

x 2
y -1
z 3
The ordered triple is (2, -1, 3)
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Lets do ANOTHER example!

• Lets solve this system of equations
• 2x y 2z 11
• 3x 2y 2z 8
• x 4y 3z 0

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Solving the system of equations continued
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Solving the system of equations continued

• Now plug in 6 for the variable z to find x and y.
  
• You should get

x 2
y -5
z 6
The ordered triple is (2, -5, 6)
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And yet ANOTHER example!

• But this one has a special twist!
• Systems of three equations can have infinite many
  solutions or no solutions.
• Here is an example with infinite many solutions

x 3y z 1 2x y 2z 2 x 2y
3z -1
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Solving a system of equations that has infinite
many solutions continued
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Summary

• Remember, solutions are written as an ordered
  triple.
• Remember, solutions can also be no solution or
  infinite many solutions.
• If one equation is missing a variable, just line
  it up with the other equations.
• For example 3p 2r 11
• q - 7r 4
• p - 6q 1
• Check your solutions in all equations to make
  sure it is correct. Just because it answers two
  of them, doesnt mean it answers all of them.

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B
A
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HW 4.4a Pg 178-179 1-27 Odd, 28-30
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4.4 Systems of Equations in Three Variables
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Groups to work on solving
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HW 4.4b Pg 178-179 2-26 Even
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HW Quiz 4.4bWednesday, December 09, 2015
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4.5 Using a System of Three Equations
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In yesterdays swim meet, Roosevelt High
dominated in the individual events, with 24
individual-event placers scoring a total of 56
points. A first-place finish scores 5 points, a
second-place finish scores 3 points, and a
third-place finish scores 1 point. Having as many
third-place finishers as first- and second-place
finishers combined really shows the teams depth.
Use a system of three equations in 3 variables
to determine the number of 1st, 2nd, and 3rd
place finishers Roosevelt had.
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You have 25 to spend on picking 21 pounds of
three different types of apples in an orchard.
The Empire apples cost 1.40 per pound, the Red
Delicious apples cost 1.10 per pound, and the
Golden Delicious apples cost 1.30 per pound. You
want twice as many Red Delicious apples as the
other two kinds combined. Write a system of
equations to represent the given information.
How many pounds of each type of apple should you
buy?
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Gina sells magazines part time. On Thursday,
Friday, and Saturday, she sold 66 worth. On
Thursday she sold 3 more than on Friday. On
Saturday she sold 6 more than on Thursday. How
much did she take in each day?
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Find a three digit positive integer such that the
sum of all three digits is 14, the tens digit is
two more than the ones digit, and if the digits
are reversed the number is unchanged.
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HW 4.5Pg 181-182 1-17 Odd, 18-19
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HW Quiz 4.5Wednesday, December 09, 2015
11 13 15 17
9 11 13 17
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4.6 Consistent and Dependent Systems
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Objective Determine whether a system of
equations is consistent or inconsistent.
Consistent System If a system of equations has
at least one solution
Inconsistent System If a system of equations
has no solution
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Objective Determine whether a system of
equations is consistent or inconsistent.
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Objective Determine whether a system of
equations is dependent
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1. Is it possible to have a system that is
   consistent and dependent?
2. Is it possible to have a system that is
   inconsistent and dependent?
3. Is it possible to have an inconsistent system
   that is not dependent?
4. Is it possible to have an consistent system that
   is not dependent?

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Objective Determine whether a system of
equations is dependent
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Determine if the systems are consistent,
inconsistent and dependent
B
A
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HW 4.6Pg 186-187 1-23 Odd, 25-31
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HW Quiz 4.6Wednesday, December 09, 2015
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4.7 Systems of Linear Inequalities
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Objective Graph a linear inequality
The boundary line of the inequality divides the
coordinate plane into two half-planes a shaded
region containing the points that are solutions
of the inequality, and an unshaded region which
contains the points that are not.
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Objective Graph a linear inequality
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Objective Graph a system of linear inequalities
A system of linear inequalities is two or more
linear inequalities in the same variables and is
also called a system of inequalities.
A solution of a system of linear inequalities is
an ordered pair that is a solution of each
inequality in the system.
The graph of a system of linear inequalities is
the graph of all solutions of the system.
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Objective Graph a system of linear inequalities
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Objective Graph a system of linear inequalities
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Objective Graph a system of linear inequalities
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Objective Graph a system of linear inequalities
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HW 4.7Pg 192 1-39 Odd, 40-41
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4.8 Using Linear Programming
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Objective Solve problems using linear
programming.
A bakery is making whole-wheat bread and apple
bran muffins. For each batch of bread they make
35 profit. For each batch of muffins they make
10 profit. The bread takes 4 hours to prepare
and 1 hour to bake. The muffins take 0.5 hour to
prepare and 0.5 hour to bake. The maximum
preparation time available is 16 hours. The
maximum baking time available is 10 hours. How
many batches of bread and muffins should be made
to maximize profits?
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Objective Solve problems using linear
programming.
Optimization means finding the maximum or minimum
value of some quantity.
Linear programming is the process of optimizing a
linear objective function subject to a system of
linear inequalities called constraints.
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Objective Solve problems using linear
programming.
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Objective Solve problems using linear
programming.
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Objective Solve problems using linear
programming.
A bakery is making whole-wheat bread and apple
bran muffins. For each batch of bread they make
35 profit. For each batch of muffins they make
10 profit. The bread takes 4 hours to prepare
and 1 hour to bake. The muffins take 0.5 hour to
prepare and 0.5 hour to bake. The maximum
preparation time available is 16 hours. The
maximum baking time available is 10 hours. How
many batches of bread and muffins should be made
to maximize profits?
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Objective Solve problems using linear
programming.
Bread
(0, 4)
(16, 2)
(0, 0)
(20, 0)
Muffins
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Objective Solve problems using linear
programming.
Wheels Inc. makes mopeds and bicycles. Experience
shows they must produce at least 10 mopeds. The
factory can produce at most 60 mopeds per month.
The profit on a moped is 134 and on a bicycle,
20. They can make at most 120 units combined.
How many of each should they make per month to
maximize profit?
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Objective Solve problems using linear
programming.
(60, 60)
(0, 60)
Mopeds
(110, 10)
(0, 10)
Bikes
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Objective Solve problems using linear
programming.
Farm Management A farmer has 70 acres of land
available for planting either soybeans or wheat.
The cost of preparing the soil, the workdays
required, and the expected profit per acre
planted for each type of crop are given in the
following table
The farmer cannot spend more than 1800 in
preparation costs nor use more than a total of
120 workdays. How many acres of each crop should
be planted to maximize the profit? What is the
maximum profit?
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(0. 30)
Wheat
(24. 12)
(0. 0)
(30. 0)
Soy
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A manufacturer of skis produces two types
downhill and cross- country. Use the following
table to determine how many of each kind of ski
should be produced to achieve a maximum profit.
What is the maximum profit?
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HW 4.8 Pg 195 1-6
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HW Quiz 4.8Wednesday, December 09, 2015
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Test Review

• Part 1

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Find a and b so that the system below has the
unique solution (-2, 3)
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Find p and q so that the graph of the equation y
x2 px q passes through (-1, 3) and (2, 4)
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Know the terms Dependent, Consistent, and
Inconsistent
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Part 2

• Part two will consist of three linear programming
  problems along with one proof

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HW R-4Pg 200 1-13 Study All Challenge Problems
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