Solving Systems Using Substitution

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Solving Systems Using Substitution

• Objective To solve systems algebraically

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Substitution Property of Equality

• If a b, then a can be used for b in any
  situation.
• Ex 1 2
• 2
• So, if were adding 1 1
• 2
• Then 2 1 3 we substituted the 2 for
  the 1
• 2 2 2
  2

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Solving for a Variable

• Solving for x
• 1st Eliminate the constant
• Divide by the coefficient
• Your answer is

• 2x 4 12
• 4 4
• 2x 16
• 2 2
• x 8

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Solving for a Variable (Part II)

• Solving for x
• 1st Eliminate the constant
• Divide by the coefficient
• Your answer is

• ax b c
• b b
• ax c b
• a a
• x c b
• a

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Systems of Equations

• Two or more equations using the same two or more
  variables.
• Ex
• 2x y 8
• y x 3
• Notice that both equations have an x and a y

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Solving a System of Equations

• Step 1 Find the variable that is already solved.
• Step 2 If no variable is solved for, find out
  which one can be solve for easier.
• Step 3 Insert the formula of the solved variable
  for the variable in the 2nd equation
• Step 4 Solve for the 2nd Variable
• Step 5 Use the answer to solve for the 1st
  variable
• Step 6 Answer must come in an ordered pair (x, y)

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Example 1

• Solve
• 2x y 8
• y x 3
• 2x (x 3) 8
• 2x x 3 8
• x 3 8
• -3 -3
• x 5

• Step 1 Find the variable that is already solved
• In this case, y is already solved for. y x 3.
  So use x 3 for y in the top equation (Step 3)
• Steps 4 Now solve for x

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Example 1 (Continued)

• x 5
• y x 3
• y 5 3
• y 2
• (5,2)

• Now plug in 5 for x in either equation. (Step 5)
• Now set answer as an ordered pair (Step 6).

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Example 2

• 3x y 13
• 4x 2y 30
• 3x y 13
• -3x -3x
• y -3x 13
• 4x 2(-3x 13) 30
• 4x 6x 26 30
• -2x 26 30
• -26 -26
• -2x 4
• -2 -2
• x -2

• Step 1 Find the variable that is already solve.
  
• Not in this case. Go to Step 2.
• To solve for x in equation 1, you need to
  subtract y and divide by 3.
• To solve for y in equation 1, you need to
  subtract 3x and thats it.
• Once you find y, plug it in on the other
  equation.
• Now solve for x
• Now that we know x, plug it in either equation to
  find y.

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Example 2 (Continued)

• y -3x 13
• y -3(-2) 13
• y 6 13
• y 19
• (-2, 19)

• Well use this equation, because it already gives
  us an answer for y.
• Our solution