Solving Equations

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Solving Equations
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Solving Equations
Equations contain an equal sign (or inequality)
and at least one variable.
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What is your Goal?

• Find what value(s) make the equation true.

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How do you accomplish it?

• Isolate the variable.
• Get rid of the furthest number that is with
  the variable first.

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Solve Isolate the variable.

• If added,
• then add the opposite
• If multiplied,
• then multiply by the reciprocal

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How do you know you are correct?

• Put your answer back into the original equation
  to see if both side have the same value.

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Questions to focus on

• Who is the variable?
• Where is the variable?
• Is the variable on the right side or left side
  of the equal sign?
• Combine like terms on the same side of the
  equation
• What numbers are with the variable?
• (on the same side of the equal sign)
• Which number is furthest away according to the
  Order of Operations? (P, E, M D, A S)

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Questions to focus on

• How is that number with the variable
• (added or multiplied)?
• How do you get rid of that number?
• (Add opposite or Multiply by reciprocal)
• Add the opposite or Multiply by the reciprocal
  to both sides of the equal sign!
• Start over with the first question until the
  variable has been isolated.

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Solve and check your answer.

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Solve and check your answer.

Check
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Solve and check your answer.

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Solve and check your answer.

Check
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Your turn to try a few problems
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Solve and check your answer. (2 operations)

• 3x 5 20

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Solve and check your answer. (2 operations)

• 3x 5 20
• 3x 5 5 20 5
• 3x 15
• 3x 15 -3 -3
• x 5

Check 3(5) 5 20 20 20
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Solve and check your answer. (2 operations)

• 7 4 6x

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Solve and check your answer. (2 operations)

• 7 4 6x
• 7 4 4 4 6x
• 3 -6x
• 3 -6x
• -6 -6
• -1/2 x

Check 7 4 6( ) 7 4 6( -1/2 ) 7 4
-3 7 4 3 7 7
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Your turn to try a few problems
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Solve and check your answer. (variables on the
same side of equation)

• 5x 2 4x 6

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Solve and check your answer. (variables on the
same side of equation)

• 5x 2 4x 6
• 5x 4x 2 6
• 1x 2 6
• x 2 2 6 2
• x 4

Check 5(4) 2 4(4) 6 20 2 16 6 22
16 6 6 6
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Solve and check your answer. (variables on both
sides of the equation)

• 4x 7 6x 2

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Solve and check your answer. (variables on both
sides of the equation)

• 4x 7 6x 2
• 4x -4x -7 6x -4x 2
• -7 2x 2
• -7 -2 2x 2 -2
• -9 2x
• -9 2x 2 2
• -9/2 x

Check 4(-9/2) 7 6(-9/2) 2 -18 7
-27 2 -18 -7 -27 2 -25 -25
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Solve and check your answer. (variables on 1
sides of the equation)

• 2x 3(x 4) 13

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Solve and check your answer. (variables on 1
sides of the equation)

• 2x 3(x 4) 13
• 2x 3(1x -4) 13
• 2x 3x -12 13
• 5x -12 13
• 5x -12 12 13 12
• 5x 25
• 5x 25 5
  5
• x 5

Check 2(5) 3((5) 4) 13 10 3(1) 13 10
3 13
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Your turn to try a few problems
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Types of solutions

• No solution, there is no value for the variable
  that will make the equation true.
• 5 2
• A unique answer, there is at least one value that
  makes the equation true.
• x number
• Infinite number of solutions, all real numbers
  makes the equation true.
• 3 3

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Solve for x.

• x 2 x 4
• x x 2 x x 4
• 2 -4
• No Solution

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Solve for x.

• 2x 1 x 5
• 2x x 1 x x 5
• x 1 5
• x 1 1 5 1
• x 6 unique solution

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Solve for x.

• x 4 4 x
• x x 4 4 x x
• 4 4
• Infinite number of solutions