Algebra Learning how to solve equations

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AlgebraLearning how to solve equations
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What is algebra all about?

• The primary purpose of algebra is to solve
  problems by using equations. Algebraic equations
  (equations used in algebra) include a variable.
  This variable represents what we are trying to
  find out the unknown.
• So we use algebra when we need to solve a problem
  using an equation.

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How could we use algebra in real life?

• People solve algebraic problems on a daily basis
  without even thinking about it. For example, if
  we know how many students go to our school and
  how many of those students are in junior high, we
  can use algebra to solve for the number of high
  school students.
• Algebra can be used to solve simple problems like
  this, but it is much more effective with slightly
  more complex problems

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How could we use algebra in real life?

• An example of a more difficult problem we can
  solve algebra for, is as follows
• Suppose your sister goes shopping with three
  weeks allowance, spends 17, and has 7 left
  over. What is her weekly allowance?
• In this example, we solve using algebra in two
  steps, which we will look at later.

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One step equations and -

• The main goal when solving equations is to get
  the variable on its own on one side of the equal
  sign.
• To do this, we have to use reverse operations to
  move all numbers to the other side of the equal
  sign.
• What ever you do, remember the golden rule for
  equations

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One step equations and -

• Whatever you do to one side of an equation, do
  also to the other side.
• Never forget this!

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One step equations and -

• Ex. x 3 8
• Our goal is to get the x by itself on the left,
  so we need to move the 3 to the other side with
  reverse operations.
• x 3 3 8 3
• Notice that we also subtracted 3 on the right,
  according to our golden rule.
• x 0 5 so x 5

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One step equations and -

• Ex. 7 x 16
• Our goal is to get the x by itself on the left,
  so we need to move the 7 to the other side with
  reverse operations. 7 is positive, so we
  subtract
• 7 7 x 16 - 7
• Notice that we also subtracted 7 on the right,
  according to our golden rule.
• 0 x 9 so x 9

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One step equations and -

• Ex. x 5 13
• Our goal is to get the x by itself on the left,
  so we need to move the 5 to the other side with
  reverse operations.
• x - 5 5 13 5
• Notice that we also added 5 on the right,
  according to our golden rule.
• x 18

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One step equations and -

• Ex. 16 x 9
• This is a special case after we move the 16,
  well be left with - x. Subtract 16 on both
  sides.
• 16 16 x 9 16
• Now we have - x - 7 We can cancel
  the negatives (both sides can be divided by (-1)
• x 7

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One step equations and -

• Ex. 4 x 10
• Move the 4 to the left by subracting on both
  sides.
• 4 4 x 10 - 4
• Now we have - x 6 We cant be left
  with a negative variable, so move the negative to
  the other side
• x - 6

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Practicing One-Step Equations Adding and
Subtracting

• Solve each of the equations. Show all steps (as
  shown in the examples).
• x 4 7 2. x 8 19
• 3. x 7 15 4. x 3 12

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Practicing One-Step Equations Adding and
Subtracting

• 5. 5 x 7 6. 13 x 19
• 7. 2 x 15 8. 6 x 12
• 9. 1 x 0 10. 53 x 97

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Practicing One-Step Equations Adding and
Subtracting

• 17. 8 - x 7 18. 4 - x 1
• 19. 9 - x 2 20. 12 - x 5
• 21. 14 - x 4 22. 40 - x 24

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Practicing One-Step Equations Adding and
Subtracting

• 23. x - 4 7 24. 5 x 8
• 25. 15 - x 11 26. x 15 23
• 27. 16 - x 3 28. x 13 3

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One step equations ? and ?

• Solving equations with ? and ? works much the
  same way as and - . We still aim to get the
  variable by itself on one side by using reverse
  operations.
• So if we are ?ing, we ? to get the variable by
  itself and vice versa.
• Note, if we multiply, we put the variable right
  beside the number like this 5x.
• If we divide, we sometimes express as a fraction
  like this

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One step equations ? and ?

• Ex. 4x 8
• Our goal is to get the x by itself on the left,
  so we need to move the 4 to the other side with
  reverse operations.
• Notice that we also divided by 4 on the right,
  according to our golden rule.
• 1x 2 so x 2

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One step equations ? and ?

• Ex. 4
• Our goal is to get the x by itself on the left,
  so we need to move the 3 to the other side with
  reverse operations.
• ? 3 4 ? 3
• Notice that we also multiplied by 3 on the
  right, according to our golden rule.
• x ? 1 12 so x 12

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One step equations ? and ?

• Ex. 2
• This is a special case where we have to move the
  variable instead. We will multiply each side by x
  first ? x 2 ? x
• Now the xs cancel on the left and we are left
  with
• 8 ? 1 2x so 8 2x
• Now we can solve by ?ing both sides by 2 .
• so 4 1 ? x so 4 x

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Practicing One-Step Equations Multiplying and
Dividing

• Solve each of the equations. Show all steps (as
  shown in the examples).
• 29. 4x 8 30. 6x 18
• 31. 3x 15 32. 2x 12

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Practicing One-Step Equations Multiplying and
Dividing

• 33. 5x 25 34. 1x 19
• 35. 7x 63 36. 6x 24
• 37. 15x 0 38. 2x 100

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Practicing One-Step Equations Multiplying and
Dividing

• 39. 2 40. 4
• 41. 5 42. 23
• 43. 0 44. 20

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Practicing One-Step Equations Multiplying and
Dividing

• 45. 2 46. 4
• 47. 5 48. 13
• 49. 8 50. 3

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Verifying answers to equations

• Whenever you solve an equation, you should check
  your answer. Today we will learn an easy way to
  do this.
• Ex. Lets say we discovered that for the
  equation x 3 10, x 7Now we use the left
  side right side method to check our answer

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Verifying answers to equations

• Left Side Right Side
• x 3 10
• (7) 3 10
• 10 10
• Since the left and right sides are equal, we know
  that our solution, x 7, is correct.

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Verifying answers to equations

• Is x 16 the solution for 5 ?
• Left Side Right Side

5
5

5.3125
5
The left side and right side are not equal, so x
? 16
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Practice verifying answers to equations

• For the numbers listed below, verify your answer
  using the left side right side method of the
  answers you found so far this unit. Example for
  1 below, verfiy your answer to 1 that you did
  earlier this unit (x 4 7)
• 1. 6.
• 12. 18.

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Practice verifying answers to equations

• 22. 31.
• 36. 41.
• 46. 50.

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Solving two-step equations

• So far weve solved equations with one operation
  , -, ?, or ?.
• Now were going to solve equations that have two
  operations.
• We can do this in the same way weve done so far,
  but we have to add another step.

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Solving two-step equations

• When we solve these two-step equations, we
  continue to move numbers to the other side of the
  equation, away from the variable.
• When there is more than one number to move, we
  work from the outside in. We start with the
  numbers not attached to the variable.
• Ex. 2a 5 13
• In this example, we first move the 5 because it
  is not attached to the variable. Then we work
  with the 2.

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Solving two-step equations

• Ex. 2a 5 13
• - 5 - 5
• 2a 0 8
• 2a 8
• 2 2
• a 4

First we subtract 5 on both sides because the 5
is not attached to the variable. Then we are
left with a simple one-step equation to solve.
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Solving two-step equations

• Ex. - 3 2
• 3 3
• 0 5
• 5 ? 4
• 1 ? b 20
• b 20

First we add 3 on both sides because the 3 is not
attached to the variable. Then we multiply both
sides by 4 to cancel the 4 on the left, and leave
the variable by itself.
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Solving two-step equations

• There are two ways to solve this equation
• 1st method
• Ex. 19 3c 13
• - 19 - 19
• - 3c - 6
• - 3 - 3
• c 2

In this method, we subtract both sides by 19
because it is not attached to the variable. Then
we are left with -3c -6. Now we have to divide
both sides by 3. Remember that a negative
divided by a negative leaves a positive!
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Solving two-step equations

• 2nd method
• Ex. 19 3c 13
• 3c 3c
• 19 0 13 3c
• - 13 - 13
• 6 3c
• 3 3
• 2 c

In this method, we move the 3c over to the right
so that we dont have a negative variable. Then
we subtract 13 on both sides so that we are left
with 6 3c. Then we can easily solve for c.
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Solving two-step equations

• We will solve these together in class
• 1. 3d 5 23 2. 2 - 1

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Solving two-step equations

• 3. 14 2f 8 4. 4 7

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Verifying answers to two-step equations

• We use the same method that we used for one-step
  equations. Now that we have two steps, remember
  to use order of operations (BEDMAS).
• The following are a few examples (the solution
  found is in brackets)
• Ex 1 2a 6 -4 (1)
• Ex 2 4b 7 -21 (-6)
• Ex 3 10 -1 (-66)

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Verifying answers to two-step equations Example 1

• a 1 is the correct answer.

Left Side 2a - 6 2(1) 6 2 6 -4
Right Side -4 -4 -4 -4
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Verifying answers to two-step equations Example 3

• f -66 is NOT the correct answer.

Left Side 10 10 -11 10 -1
Right Side 1 1 1 1
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Practice verifying answers to two-step equations

• Verify answers on a separate piece of paper using
  the left side right side method, for the
  following questions which you completed on a
  separate piece of paper
• s 3, 8, 14, 22, 28, 31, 34, 39, 41, 46

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Problem solving with equations

• When you are trying to solve a word problem, you
  should do 4 steps to help you
• Step 1 Define the variable
• Step 2 Set up the equation
• Step 3 Solve for the variable and make sure
  your answer makes sense
• Step 4 State the solution in a sentence

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Problem solving with equations

• Ex. Ellen works for McDonalds and gets paid
  8.00 per hour. She wants to buy a snowboard
  that costs 304.95, and she has already saved
  155 towards this cost. How many more hours will
  she have to work until she has enough money for
  the snowboard?
• Step 1 Define the variable.
• We are trying to find the number of hours she
  needs to work, so let the number of hours be h.

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Problem solving with equations

• Ex. continued
• Step 2 Set up the equation.
• Total saved must equal 304.95.
• Total saved must be what she has saved already
  plus the money she will make from working.
• So 155 money from working 304.95
• Money from working equals 8.00 times the number
  of hours , which is h.
• So 155 8h 304.95

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Problem solving with equations

• Ex. continued
• Step 3 Solve for the variable and ask yourself
  whether it makes sense.
• 155 8h 304.95
• - 155 - 155
• 8h 149.95
• 8 8
• h 18.74
• This sounds about right, but she cant work only
  part of an hour, so lets say 19 hours.
• Step 4 Sentence Ellen must work 19 more hours
  until she can buy the snowboard.

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Practice solving problems with equations

• The following problems use one-step equations.
  Using the 4 steps, solve at least 6 questions on
  a separate piece of paper and put in your
  workbook.
• 1. Alicia paid 4.85 for a sandwich and juice.
  If the price of the sandwich was 3.50, how much
  was the juice?
• 2. Jonathan is paid 8.40 per hour. How much
  will he be paid if he works 5 and a half hours?

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Practice solving problems with equations

• A can of chili costs 2.79. How many whole cans
  of chili can you buy with twenty dollars?
• Sarah is a waitress at Boston Pizza. She just got
  a raise of 1.20 per hour and now makes 9.15 per
  hour. What was her wage before the raise?
• Mark wants to buy his mother a microwave for her
  birthday. The cost of the microwave is 174.39.
  If Mark makes 8.75 per hour, how many full hours
  does Mark have to work to save up for the
  microwave?

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Practice solving problems with equations

• A recipe for apple pie requires 4 ½ cups of
  apples. If 2 ¾ cups are in the mixing bowl, how
  many more cups of apples are needed?
• Loaves of bread are advertised at a price of
  three for 2.79. How much is this per loaf?
• Lauras parents took her out to dinner on her
  sixteenth birthday. The amount of the bill was
  102.85. Her parents left 120. How much was
  the tip?

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Practice solving problems with equations

• The following problems use two-step equations.
  Using the 4 steps, solve at least 6 questions
  again on a separate piece of paper and put in
  your workbook.
• 1. The school volleyball team went through a
  drive-thru fast food restaurant on the way home
  from a tournament. They ordered nine milkshakes
  and twelve hamburgers. The manager paid the bill
  of 42.21. They players tried to figure out how
  much they owed, and one person remembered that
  the cost of one hamburger was 2.10. What was
  the price of one milkshake?

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Practice solving problems with equations

• Carl, James, and Rob decided to celebrate the end
  of the semester by going out for dinner. At the
  end of the meal, they each contributed an equal
  amount to the cost. Then they realized they
  forgot the tip, so they added an extra 6.00. If
  the entire cost of the dinner, including the tip,
  was 45, how much did each boy pay before adding
  the tip?
• On her last visit to the grocery store, Larissa
  spent half the money she had on fruits and
  vegetables. She also bought a loaf of bread for
  1.39. If she had 2.36 left after paying for
  her groceries, how much money did she have when
  she entered the store?

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Practice solving problems with equations

• During a recent trip to the grocery store, Brian
  bought 2 dozen eggs and a litre of milk. His
  total came to 3.87. If a litre of milk cost
  1.49, what was the price of one dozen eggs?
• Tamara would like to buy a new pair of jeans
  which cost 74.90, tax included, and she has
  already saved 35 toward their cost. Her job as
  a server pays 7.65 per hour. How many hours
  will Tamara have to work to buy the jeans?
• Kevin works as a server at a busy family
  restaurant. Last Saturday he spent one-sixth of
  his tips on dinner after his shift. The cab ride
  home was 3.75, which he also paid from his tips
  that day. The total amount he spent was 11.75.
  How much did he make in tips?

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Practice solving problems with equations

• Jadin baked three kinds of cookies chocolate
  chip, oatmeal raisin, and peanut butter. He made
  equal numbers of chocolate chip and peanut butter
  cookies. He also made thirty oatmeal raisin
  cookies. He baked a total of 110 cookies. How
  many peanut butter cookies did he make?
• The best player on the Mullets hockey team scored
  30 goals last season, which was one third of the
  total goals scored for the team. If the Mullets
  total goals scored last season was 15 less than
  their rivals, the Skullets, how many did the
  Skullets score?

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Practice solving problems with equations

• The Mullets won the local hockey championship and
  celebrated by ordering pizza pepperoni and
  hawaiian. Pepperoni pizzas cost 10.00 each and
  a third of the bill paid for the hawaiian pizzas.
  If the total amount of the bill was 60.00, how
  many pepperoni pizzas were ordered?
• For a school event, cans of pop were ordered.
  Each can cost 0.63, and all unopened cans could
  be returned for a full refund. The final amount
  paid was 272.16 after 18 unopened cans were
  returned. How many cans did the school order in
  total?