Solving Systems of Equations

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Section 2-1

• Solving Systems of Equations

2
Different Types of Systems

• 1.) Consistent a system that has at least one
  solution
• a.) Independent exactly one solution, the
  lines intersect
• b.) Dependent has an infinite number of
  solutions, equations represent the same line

2.) Inconsistent a system with no solution,
the lines do not intersect (parallel lines)
3
Determine what type of system

• 2x 5y 8
• 3x 4y 5

m1 2/5
m2 -3/4
Since the slopes are not equal, the lines will
intersect. There is one solution.
Therefore, we say the system is Consistent and
Independent. (C I)
4
Determine what type of system

• 3x 3y 12
• x y 4

m1 -1
b 4
b 4
m2 -1
Since the slopes and the intercepts are equal, it
is the same line. So we have infinite solutions.
Therefore, we say the system is Consistent and
Dependent. (C D)
5
Determine what type of system

• 4x 2y 10
• 2x y 10

m1 2
b1 -5
m2 2
b2 -10
Since the slopes are the same and the intercepts
are different, it is two parallel lines. So we
have no solution.
Therefore, we say the system is Inconsistent.
(Inc.)
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Solving Algebraically

• Solve by substitution
• 3x 2y 7
• x y 4

y 4 x

• 3x 2 (4 x) 7
• 3x 8 2x 7
• 5x 15
• x 3
• y 1

(3, 1)
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Solving Algebraically

• Solve using elimination
• 3x 2y 7
• x y 4
• 2 (x y) 4 (2)
• 2x 2y 8
• 5x 15
• x 3
• y 1

Add the equations
(3, 1)
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Solve by Elimination
6x 10y 40 20x 10y 90

• 3x 5y 20
• 4x 2y 18

26x 130 x 53(5) 5y 20 -5y
5 y -1 (5, -1)
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Solve by Graphing

• 3x 5y 20
• 4x 2y 18

y 3/5x 4 y -2x 9
(5, -1)
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Application

• 48 CDs and 32 cassettes cost 1040
• 3 times as many CDs and 4 times as many
  cassettes cost 3440.
• 48x 32y 1040
• 144x 128y 3440
• Solve.
• x cost of CDs
• y cost of cassettes

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Solve by Elimination

• 48x 32y 1040
• 144x 128y 3440
• (-3)(48x 32y) 1040(-3)
• -144x 96y -3120
• 32y 320
• y 10
• 48x 32(10) 1040
• 48x 720
• x 15

CD 15Cassettes 10
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Break Even Point

• The cost function for producing a widget is y
  3x 80.
• The revenue function is y 8x.
• Solve to find the break even point.
• 3x 80 8x
• 30 5x
• 6 x
• What does it mean to break even?
• Graph it!

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Homework

• Page 60
• 13-15, 17-27odd, 30-32, 35