Solving Equations

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Solving Equations

• by Lauren McCluskey

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Credits

• Prentice Hall Algebra I

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Solving One-Step Equations

• An equation is like a balance scale because it
  shows that two quantities are equal. The scales
  remained balanced when the same weight is added
  (or removed from) to each side.
• x 2 5 x (-5) -2

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What does it mean to Solve an equation?

• To solve an equation containing a variable, you
  find the value (or values) of the variable that
  make the equation true.
• Get the variable alone on one side of the equal
  signusing inverse operations, which are
  operations that undo each other.

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Inverse Operations

• Addition and Subtraction are inverse operations
  because they undo each other.
• Multiplication and division are inverse
  operations because they undo each other.

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Properties of Equality

• Addition Property of Equality For every real
  number a, b, and c, if a b, then a c b
  c.
• Subtraction Property of Equality For every real
  number a, b, and c, if a b, then a - c b -
  c.

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Properties of Equality

• Multiplication Property of Equality For every
  real number a, b, and c, if a b, then ac
  bc.
• Division Property of Equality For every real
  number a, b, and c, if a b, then a/c b/c.

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Using Reciprocals
2/3x 12

• In order to solve the equation above, you need to
  divide by 2/3. Remember To divide a fraction,
  you multiply by its reciprocal. In other words
  flip it!

3/22/3x 1x 3/2 12/1 18 So x 18.
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Examples
-14

1. a - 4 -18
2. b 24 19
3. v/ 3 -4
4. 15c 90
5. -1/5 r -4

-5
-12
6
20
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Try It!

1. d (-4) -7
2. -54 q - 9
3. m / -4 13
4. -75 -15x
5. 2/3 n 14

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Check your answers

• -3
• -45
• -52
• 7
• 21

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Solving Two-Step Equations

• A two-step equation is an equation that involves
  two operations.
• PEMDAS tells us to multiply or divide before we
  add or subtract, but to solve equations, we do
  just the opposite we add or subtract before we
  multiply or divide.

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Try It!

• 7 2y - 3
• 6a 2 -8
• 3) x/9 - 15 12
• 4) -x 7 12
• 5) a -5 -8
• 6) 4 -c 11

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Check your answers

• 7 2y - 3
• 3 3
• 10 2y
• 2 2
• 5 y
• 2) 6a 2 - 8
• -2 - 2
• 6a -10
• 6 6
• a -1 2/3

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• To Solve Multi-Step Equations
• Clear the equation of fractions and decimals.
• Apply the Distributive Property as needed.
• Combine like terms.
• Undo addition and subtraction.
• Undo multiplication and division.

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Example

• 1/5 3w/ 15 4/5
• To clear fractions every term by 5
• 5 1/5 1 5 3w/15 9w 5 4/5 4
• So 1 9w 4
• Undo addition
• -1 -1
• 9w 3
• Undo multiplication
• 9w / 9 3 / 9
• So w 1/3

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Try It!

1. a/7 - 5/7 6/7
2. 9y/ 14 3/7 9/14
3. 2/3 3/k 71/12

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Check your answers

1. a 11
2. y 1/3
3. k 4/7

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Example

• 0.11p 1.5 2.49
• Clear decimals by by 100
• 100 0.11 11 100 1.5 150 100 2.49 249
• So 11p 150 249
• Undo addition
• -150 -150
• 11p 99
• Undo multiplication
• 11p/11 99/11
• p 9

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Try it!

1. 25.24 5y 3.89
2. 0.25m 0.1m 9.8
3. 26.54 - p 0.5(50 - p)

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Check your answers

• y 4.27
• m 28
• p 3.08

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Try it!

1. -4(x 6) -40
2. m 5(m -1) 7
3. 1/4(m - 16 ) 7

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Check your answers

1. x 4
2. m 2
3. m 12

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Equations with Variables on Both Sides

• Use the Addition or Subtraction property of
  Equality to get the variables on one side of the
  equation.

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Example

• 4p - 10 p 3p -2p
• Combine like terms
• p 3p - 2p 2p
• Use the subtraction property of equality
• 4p - 10 2p
• -2p -2p
• 2p -10 0

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Example cont.

• 2p - 10 0
• Undo subtraction
• 10 10
• 2p 10
• Undo multiplication
• 2p / 2 10/ 2
• p 5

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Try It!

1. 6b 14 -7 - b
2. -36 2w -8w w
3. 30 - 7z 10z - 4

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Check your answers

1. b -3
2. w 4
3. z 2

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Identity or No Solution

• An equation has no solution if no value of the
  variable makes the equation true.
• An equation that is true for every value of the
  variable is an identity.

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2.5 Defining One Variable in Terms of Another

• Some problems involve two or more unknown
  quantities. To solve such problems, first decide
  which unknown quantity the variable will
  represent. Then express the other unknown
  quantity in terms of that variable.

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Example

• The width of a rectangle is 2 cm less than its
  length. The perimeter of the rectangle is 16cm.
  What is the length of the rectangle?
• Let l length
• Let l - 2 width
• P 2l 2w
• So 2(l) 2(l -2) 16cm

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Example cont.

• 2l 2(l-2) 16 cm
• 2l 2l - 4 16cm
• 4l - 4 16 cm
• 4 4
• 4l 20 cm
• 4l / 4 20/ 4 so l 5

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Try it!

• The length of a rectangle is 6 more than 3 times
  as long as the width. The perimeter is 36 m. What
  are the measurements?

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Check your answer

• l 15m
• w 3m
• OR 15m x 3m

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Consecutive Integers

• Consecutive integers differ by 1. Consecutive
  even or consecutive odd integers differ by 2.
• Example
• The sum or two consecutive odd integers is 84.
  What are the integers?

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Consecutive Integers

• Let x the 1st integer
• Let x 2 the 2nd integer
• x x 2 84
• 2x 2 84
• -2 -2
• 2x 82
• x 41 x 2 43
• So the integers are 41 and 43.

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Try It!

• The sum of three consecutive integers is 48. What
  are the integers?

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Check your answer

• The integers are
• 15, 16, and 17.

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Rate Times Distance

• When the distances covered are equal, we can set
  the two expressions equal to each other and solve
  for x.
• When the distances combine to make up the total
  distance, we can add the expressions, set it
  equal to the total distance, and solve for x.

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Try It!

• Adapted from Prentice Hall
• 1) A group of campers left the campsite in a
  canoe going 10km/h. Two hours later, another
  group left in a motor boat going 22km/h. How long
  did it take the second group to catch up?

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Try It!

• 2) On his way to work, your uncle averaged 20
  mph. On his way home, he averaged 40mph. If the
  total time was 1 1/2hours, how long did it take
  him to drive to work?

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Try It!

• 3) Sarah and John left Perryville going in
  opposite directions. Sarah drives 12mph faster
  than John. After 2 hours, they are 176 miles
  apart. Find Sarahs and Johns speeds.

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Check your answers

• 1 3/4 hours
• 1 hour
• John 33mph
• Sarah 45mph