Graphing Systems of Equations

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Chapter 8
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Section 8.1
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Graphing Systems of Equations

• Graph each of the equations.
• The solutions of the system are given by the
  points of intersection.
• Any system with exactly one solution is said to
  be independent.
• If the graphs do not intersect, then there are no
  solutions and we say the equations are
  inconsistent.
• If the graphs are identical, then we say every
  point on the graph is a solution and we say the
  equations are dependent.

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Section 8.2
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The Substitution Method

• Given a system of two equations in the variables
  x and y.
• Solve one of the equations for y.
• Substitute this expression into the other
  equation in place of y then solve the resulting
  equation for x.
• Go back to one of the original equations and use
  this value for x to solve for the variable y.
• Check the solution.

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Problem Solving

• Analyze the problem.
• Define your variables and form a system of
  equations.
• Solve the system.
• Check the solution.
• State the conclusion.

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A problem from ancient China (152 BC)
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A problem from Wonderland
In Lewis Carrolls Through the Looking Glass,
Tweedledum says to Tweedledee, The sum of your
weight and twice mine is 361 pounds. Then
Tweedledee says to Tweedledum, Contrariwise, the
sum of your weight and twice mine is 362 pounds.
Find the weight of each.
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Example
A preschool charges 8 per child to attend the
morning session, and 10 to attend the afternoon
session. No child can attend both. Thirty
children are enrolled in the preschool. If the
daily receipts are 264, how many children attend
each session?
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Example
A salesman receives a commission of 3 for every
pair of dress shoes he sells. He receives a
commission of 2 for every pair of athletic shoes
he sells. After selling 9 pairs of shoes in one
day, his commission was 24. How many pairs of
each type of shoe did he sell?
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The Elimination Method

• Given a system of two equations in the variables
  x and y.
• Rewrite each equation in the form AxByC
• Add a multiple of one equation to the other
  equation so that one of the variables is
  eliminated.
• Solve the resulting equation for the variable.
• Go back to one of the original equations and use
  this value to solve for the other variable.
• Check the solution.

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Section 8.3
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Problem Solving

• Analyze the problem.
• Define your variables and form a system of
  equations.
• Solve the system.
• Check the solution.
• State the conclusion.

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Example
For every problem answered correctly on an exam,
3 points are awarded. For every incorrect answer,
4 point are deducted. In a 10 question test, a
student scored 16 points. How many correct and
incorrect answers did the student have on the
exam.
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Example
A part-time movers regular pay rate is 6 per
hour. If the work involves going up and down
stairs, his rate increases to 9 per hour. In one
week he earned 138 and worked 20 hours. How many
hours did he work at each rate?
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Example
A fitness club has 150 members. Monthly
membership fees are 25 for those under 65 and a
special senior rate of 15 for those 65 and
older. If there are twice as many seniors as
regular paying customers, how much does the club
receive each month?
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Mixture Problems
Sunflower seed costs 1.00 per pound. Rolled oats
cost 1.35 per pound. How many pounds of each
seed would you need to make 50 lbs of a mixture
that costs 1.14 per pound?
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Example
A caterer needs to provide 10 lbs of mixed nuts
for a wedding reception. Peanuts cost 2.50 per
pound and fancy nuts cost 7.00 per pound. If 40
has been budgeted for nuts, how many pounds of
each can be used?
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Example
A collection of dimes and quarters is worth
15.25. There are 103 coins in all. How many
dimes and how many quarters are there?
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Example
A solution containing 28 fungicide is to be
mixed with a solution containing 40 fungicide to
make 300 liters of a solution containing 36
fungicide. How much of each solution is needed?
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Example
A jeweler wishes to make a 60 oz mixture that is
two-thirds pure gold. She has two stocks of gold
alloy, the first stock contains three-fourths
pure gold and the second stock is five-twelfths
pure gold. How many ounces of each stock does she
need?
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Graphing Linear Inequalities

• Graph the corresponding equation.
• Use a dashed line if there is a lt or gt .
• Use a solid line if there is a lt or gt .
• Check one ordered pair to determine which side of
  the line to shade.
• If possible use the ordered pair (0,0).

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