Solving Systems of Equations Algebraically

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Solving Systems of Equations Algebraically

• Chapter 3.2

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Alternatives to Graphing

• Sometimes, graphing systems is not the best way
  to go
• Lines dont intersect at a discernable point
• Dont have enough room to graph
• Etc.
• Alternate methods of solving systems
• Substitution
• Elimination

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Substitution

• Substitution
• One equation is solved for one variable in terms
  of the other. This expression can be substituted
  for the variable in the other equation.

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Example 1

• x 4y 26
• x - 5y -10
• x 5y 10
• 5y 10 4y 26
• 9y 10 26
• 9y 36
• y 4

• Solve either equation for one variable
• Substitute this expression into the other
  equation
• Now, there should only be one variable
• After solving for one variable, plug it into
  either equation to solve for other variable

x 4y 26 x 4(4) 26 x 16 26 x
10 Solution (10, 4)
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Example 2

• 2x y 4
• 3x 2y 1
• y -2x 4
• 3x 2(-2x 4) 1
• 3x - 4x 8 1
• -x 8 1
• -x -7
• x 7

• Solve either equation for one variable
• Substitute this expression into the other
  equation
• Now, there should only be one variable
• After solving for one variable, plug it into
  either equation to solve for other variable

2x y 4 2(7) y 4 14 y 4 y
-10 Solution (7, -10)
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Elimination

• Elimination
• Eliminate one of the variables by adding or
  subtracting the two equations together

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Example 3
x y 6 x 4 6 x 2 Solution (2, 4)
x 2y 10 x y 6 0 y 4 y 4

• Instead of using Substitution, you can subtract
  one equation from the other
• Subtract the 2nd from the 1st
• After solving for one variable, plug it into
  either equation to find the other

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Example 4
2x y 5 2(5) y 5 10 y 5 y
-5 Solution (5, -5)
2x y 5 3x - y 20 5x 0 25 x 5

• Instead of using Substitution, you can add one
  equation to the other
• After solving for one variable, plug it into
  either equation to find the other

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Elimination with Multiplication

• When one variable cannot be easily eliminated
  using simple addition or subtraction, multiply
  one or both equations by constants so that a
  variable CAN be eliminated.

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Example 5
2x 3y 12 2x 3(2) 12 2x 6 12 2x 6 x
3 Solution (3, 2)

• Simple addition or subtraction isnt going to
  help here.
• Decide which variable to eliminate
• Lets do x
• Now subtract 2nd from 1st
• Use one variable to solve for the other

2x 3y 12 5x - 2y 11 10x 15y 60 10x -
4y 22 0 19y 38 y 2
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Example 6
2g h 6 2(4) h 6 8 h 6 h
-2 Solution (4, -2)

• Simple addition or subtraction isnt going to
  help here.
• Decide which variable to eliminate
• Lets do h
• Now add together
• Use one variable to solve for the other

2g h 6 3g - 2h 16 4g 2h 12 3g - 2h
16 7g 0 28 g 4
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Inconsistent and Dependent Systems

• If you add or subtract two equations in a system
  and the result is an equation that is never true,
  then the system is inconsistent and it has no
  solution.
• Examples
• 1 2
• -1 1
• If the result is an equation that is always true,
  then the system is dependent and has infinitely
  many solutions.
• Examples
• 1 1
• 9 9

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Example 7 (substitution)

• x 3y 8
• 1/3 x y 9
• x -3y 8
• 1/3(-3y 8) y 9
• -y 8/3 y 9
• 8/3 9
• Not True

• Solve either equation for one variable
• Substitute this expression into the other
  equation
• Now, there should only be one variable

No Solutions!!
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Example 8 (substitution)

• 2a - 4b 6
• -a 2b -3
• a 2b 3
• 2(2b 3) - 4b 6
• 4b 6 - 4b 6
• 6 6
• Always True

• Solve either equation for one variable
• Substitute this expression into the other
  equation
• Now, there should only be one variable

Infinitely Many Solutions!!
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Example 9 (elimination)

• Simple addition or subtraction isnt going to
  help here.
• Decide which variable to eliminate
• Lets do x
• Now add together

4x - 2y 5 -2x y 1 -4x 2y 2 4x - 2y
5 0 0 7 Not True
No Solutions