Linear Equations

1
Linear Equations
2
Linear Equations

• Solving linear equations
• Whatever you do to one side of an equation, you
  must also do to the other.
• Goal is to get the variable alone on one side of
  the equation by undoing the listed operations.

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Linear Equations

• Examples
• X 4 8
• 4 4 subtract 4 (or add 4) to
  both sides
• X 4
• To check substitute your solution for the
  variable and see if the statement is true.
• X 4 8, X 4 therefore
• 4 4 8 (true)

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Linear Equations

• 4x 48
• 4 4 since 4 is multiplying x we
  divide both
• sides by 4 to make the
  coefficient of x 1
• x 12
• Simplify first to get only one term with the
  variable!
• 2x 3x 13 2
• 5x 15 simplify by combining
  like terms
• 5 5 divide both sides by 5
• x 3

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Linear Equations

• Two operations
• 2x 4 8 nothing to simplify
• 4 4 subtract 4 from both sides
• 2x 4
• 2 2 divide both sides by 2
• x 2

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Linear Equations

• Variables on both sides of equation
• 9 4x 8x cant simplify since 4x and 8x are
• on opposite sides of the equation. Remember
  the goal is to get one term with the variable!
• 9 4x 8x
• 4x 4x add 4x to both sides to
  get the x
• terms together
• 9 12x
• 12 12 divide both sides by 12
• ¾ x

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Linear Equations

• Variables on both sides (2 methods)
• 9x 2 6x 10 Method 1 move constants first
• 2 2 Subtract 2 from both
  sides
• 9x 6x 12
• 6x 6x subtract 6x from both sides
• 3x 12
• 3 3 divide both sides by 3
• x 4

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Linear Equations

• Variables on both sides (2 methods)
• 9x 2 6x 10 Method 2 move variables
  first
• 6x 6x subtract 6x from both
  sides
• 3x 2 10
• 2 2 subtract 2 from both
  sides
• 3x 12
• 3 3 divide both sides
  by 3
• x 4

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Linear Equations

• Applying the Distributive Property
• 28 7x 3(x 4)
• 28 7x ( 3)x ( 4) change subtract to
  add opp.
• 28 7x ( 3x) 12 distribute the 3 to
  all terms
• 28 4x 12 simplify
• 12 12 subtract 12
  from both sides
• 16 4x
• 4 4 divide both
  sides by 4
• 4 x

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Linear Equations

• Applying the Distributive Property
• 6 ( x 3 ) 2x think of the before the
  parenthesis as a 1
• 6 1(x 3) 2x
• 6 1x 3 2x distribute the 1 thru the
  parenthesis
• 3 1x 2x simplify
• 1x 1x add 1x to both sides
• 3 3x
• 3 3 divide both sides by 3
• 1 x

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Linear Equations

• Special cases (1)
• 4x 3 3x 3 x
• 4x 3 4x 3 simplify
• 4x 4x subtract 4x from both sides
• 3 3
• Since 3 DOES equal 3, we say that ALL real
  numbers are solutions!

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Linear Equations

• Special cases (2)
• 3x 7 x 2x 11
• 3x 7 3x 11 simplify
• 3x 3x subtract 3x from both sides
• 7 11
• Since 7 DOES NOT equal 11, we say that there
  is NO solution to this equation. In other words,
  no real number will make this equation true!