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SLIDE 1
Formulas use letters as symbols to represent numbers
We use formulas by replacing the letters by numbers
Find V when and
SLIDE 2
When and
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The formula comes from Physics. It is used to express the relationship between voltage (V), current (I) and resistance (R).
Physical quantities have units of measurement. For example, voltage is measured in volts, current in amps and resistance in Ohms.
Here we ignore the units, unless they are units of measurement of length, area, volume, mass and the like. So, we are leaving out the units and just working with the formulas.
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Find I when
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Find P when
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Find v when
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When
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When
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When
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In a formula like
we find the use of the multiplication symbol a little distracting. It “gets in the way”. So, we have a rule that when letters are written next to each other (juxtaposed), that means that they represent numbers that are multiplied together
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Find p when
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Find E when
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Find E when
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When
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When
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When
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In the formula
The symbol I is called the subject of the formula. This is because it is written alone, to the left, and all the other letters are stated to be equal to it.
The subject could be written on the right-hand side too, but in English we read from left to right, and we usually place the subject of a formula on the left. This is for convenience and presentation only. Equations are the same, whichever way around they are written.
In each case, state the subject of the formula
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Formula Subject
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p
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E
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y
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In formulas letters represent quantities that can be measured. The use of letters is a shorthand for the lengthier word equation.
where
Write the following as word equations
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where
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where
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where
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Rewrite the following word equations using the letters indicated
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where
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where
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Example
Solve
Solve
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When you take a number from one side of an equation to the other it changes sign
Example
Solve
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When you take a number from one side of an equation to the other it changes sign
Example
Solve
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Example
Solve
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Example
Solve
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Any letter could be used
Solve
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Example
This asks for the number that when it is divided into 60 the result is 6. This number is 10.
Solve
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We can use an equation to find any of the quantities involved.
Example
Find F when
We begin by substituting all the numbers for letters
We then solve for the unknown quantity
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Find R when
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Find T when
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When
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When
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Find m when
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Find t when
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When
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When
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Collecting like terms
Collect the following
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Simplify
Simplify the following
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Two-step equations
Solve
Step 1
Step 2
Solve
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SLIDE 38
Forming expressions
A book costs x and a magazine costs y
(Both costs are in pounds.)
The cost of 5 books and 3 magazines is
Write an expression for
? The cost of six books and three magazines
? The cost of ten magazines and six books
? The difference in cost between 13 books and 6 magazines, where the cost of the books is greater than the cost of the magazines.
? The change left over from £100 when one pays for 5 books and 4 magazines
SLIDE 39
? The cost of six books and three magazines
? The cost of ten magazines and six books
This is the same as
? The difference in cost between 13 books and 6 magazines, where the cost of the books is greater than the cost of the magazines.
? The change left over from £100 when one pays for 5 books and 4 magazines
This is the same as
SLIDE 40
Write expressions for
? A girl is n years old. How old was she 5 years ago?
? In a game, a player had x ten point cards, y five point cards and z one point cards. What was his total score?
? x and y are two different numbers. What is the sum of these two numbers?
? The sum of 8 times a number x and 3 times a number z
? The product of four different numbers a, b, c and d
SLIDE 41
? A girl is n years old. How old was she 5 years ago?
? In a game, a player had x ten point cards, y five point cards and z one point cards. What was his total score?
? x and y are two different numbers. What is the sum of these two numbers?
? The sum of 8 times a number x and 3 times a number z
? The product of four different numbers a, b, c and d
SLIDE 42
? A boy’s age today is n years. Nine years ago the boy was 6 years old. What is n?
? Three fifths of a number N added to 15 is equal to 30.
Find N.
? A class had x students. After 21 students were added to the class there were students. Find x.
SLIDE 43
? A boy’s age today is n years. Nine years ago the boy was 6 years old. What is n?
? Three fifths of a number N added to 15 is equal to 30.
Find N.
? A class had x students. After 21 students were added to the class there were students. Find x.
SLIDE 44
? When 48 is subtracted from the value is 96. What is x?
? When a number n is multiplied by 0.01 the result is 0.1. What is n?
? I think of a number, k. I add 24 to k. The result is five times k. What is k?
SLIDE 45
? When 48 is subtracted from the value is 96. What is x?
? When a number n is multiplied by 0.01 the result is 0.1. What is n?
? I think of a number, k. I add 24 to k. The result is five times k. What is k?
SLIDE 46
In each case find only the positive value
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SLIDE 48
? A room is x metres square. If the room is , what is x?
? A corridor is 7 times longer than its width. If the area of the corridor is what is are its length and width?
SLIDE 49
? A room is x metres square. If the room is , what is x?
? A corridor is 7 times longer than its width. If the area of the corridor is what is are its length and width?
Let the width of the corridor be y
width = 4 m, length = 28 m
SLIDE 48B
? A room is x metres square. If the room is , what is x?
? A corridor is 7 times longer than its width. If the area of the corridor is what is are its length and width?
SLIDE 49B
? A room is x metres square. If the room is , what is x?
? A corridor is 7 times longer than its width. If the area of the corridor is what is are its length and width?
Let the width of the corridor be y
width = 4 yd, length = 28 yd
SLIDE 50
The length of a room is 5 metres longer than its width. Let the width of the room be x. The perimeter of the room is 90 m. What is x?
SLIDE 51
The length of a room is 5 metres longer than its width. Let the width of the room be x. The perimeter of the room is 90 m. What is x?
SLIDE 50B
The length of a room is 5 yards longer than its width. Let the width of the room be x. The perimeter of the room is 90 yd. What is x?
SLIDE 51B
The length of a room is 5 yards longer than its width. Let the width of the room be x. The perimeter of the room is 90 yd. What is x?
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Moving a symbol from one side of an equation to another
The algebraic symbol changes sign when it crosses the equals sign just like a number.
You then collect like terms and solve
Example
Solve
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Multiplying through by
Multiply the following through by
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You can have 0 on one side of an equation
This has happened by bringing the x to the left.
Bring x to the left
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There are sometimes several ways to reach the same solution
Solve
METHOD 1 METHOD 2
Take x to the right Take x to the left
The solution is the same whichever way you find it, but some methods are quicker than others. In this case the second method is quicker.
Solve by two methods
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SLIDE 59
Take x to the right Take x to the left
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SLIDE 60
Let the unknown number be x
Some people were in a room. 6 people entered the room by one door and 2 people left the room by another. Then there were twice as many people in the room as before. What was the number of people in the room at the beginning?
Complete the following
Let the number of people in the room at the beginning be x.
6 people entered the room
Then 2 people left the room
Double the original number
Equation
Solution
SLIDE 61
Some people were in a room. 6 people entered the room by one door and 2 people left the room by another. Then there were twice as many people in the room as before. What was the number of people in the room at the beginning?
Solution
Let the number of people in the room at the beginning be x.
6 people entered the room
Then 2 people left the room
Double the original number
Equation
Solution
SLIDE 62
At the start of the day there were some cars parked in a street. During the day 4 cars drove off, but another 10 cars were parked. By the end of the day there were 3 times as many cars as at the start. How many cars were there at the start?
Complete the following
Let the number of cars parked at the start of the day be x
4 cars drove off
10 cars parked
3 times the original number
Equation
Solution
SLIDE 63
At the start of the day there were some cars parked in a street. During the day 4 cars drove off, but another 10 cars were parked. By the end of the day there were 3 times as many cars as at the start. How many cars were there at the start?
Complete the following
Let the number of cars parked at the start of the day be x
4 cars drove off
10 cars parked
3 times the original number
Equation
Solution
SLIDE 64
Between midnight and sunrise, the temperature went down by 5°C. By noon, the temperature went up 10°C, at which time the temperature was double what it was at midnight. What was the temperature at midnight?
Hint: Let the unknown temperature at midnight be x. Form an equation, and then solve it.
SLIDE 65
Between midnight and sunrise, the temperature went down by 5°C. By noon, the temperature went up 10°C, at which time the temperature was double what it was at midnight. What was the temperature at midnight?
Solution
Temperature at midnight
Temperature at sunrise
Temperature at noon
Double the temperature
Equation
Solution
It was 5°C at midnight
SLIDE 66
At the beginning of the day there was a stock of chairs at the warehouse of the furniture store. During the day three times that number was delivered to the warehouse, and 8 chairs were removed. There were twice as many chairs in the warehouse at the end of the day than at the beginning. How many chairs were there at the beginning of the day?
SLIDE 67
At the beginning of the day there was a stock of chairs at the warehouse of the furniture store. During the day three times that number was delivered to the warehouse, and 8 chairs were removed. There were twice as many chairs in the warehouse at the end of the day than at the beginning. How many chairs were there at the beginning of the day?
Solution
SLIDE 68
As you get better and better at solving equations, you may omit lines in your working.
LONG SHORT
Solve the following in three steps
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Clearing a fraction
Multiply EVERYTHING by 3
And solve
Clear the fraction and solve
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SLIDE 72
FUNDAMENTAL RULE OF EQUATIONS
What you do to all of one side of the equation you also do to all the other
Example
Explain in each case how the fundamental rule of equations has been applied.
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FUNDAMENTAL RULE OF EQUATIONS
What you do to all of one side of the equation you also do to all the other
In each of the following this rule has not been applied and a blunder has been made. Explain what the blunder is and give the correct version.
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SLIDE 75
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SLIDE 76
There were a number of pennies in a piggy box. 20 pennies were removed, but later 5 pennies were added. By this time there were one quarter of the pennies in the piggy box than at the beginning. What was the number of pennies in the piggy box at the start?
SLIDE 77
There were a number of pennies in a piggy box. 20 pennies were removed, but later 5 pennies were added. By this time there were one quarter of the pennies in the piggy box than at the beginning. What was the number of pennies in the piggy box at the start?
Solution
SLIDE 78
There were 1p, 2p and 5p coins in a piggy box. Three 1p coins were removed, four 2p coins were added, and five 5p coins were removed. The value of the money in the piggy box at the end was one third the value at the start. Find the value of the money at the start.
SLIDE 79
There were 1p, 2p and 5p coins in a piggy box. Three 1p coins were removed, four 2p coins were added, and five 5p coins were removed. The value of the money in the piggy box at the end was one third the value at the start. Find the value of the money at the start.
Solution
Let the value at the start be x
Three 1p coins were removed, four 2p coins were added, and five 5p coins were removed ?
The value of the money in the piggy box at the end was one third the value at the start ?
SLIDE 78B
There were 1¢, 5¢ and 10¢ coins in a piggy box. Three 1¢ pennies were removed, three 5¢ nickels were added, and five 10¢ dimes were removed. The value of the money in the piggy box at the end was one third the value at the start. Find the value of the money at the start.
SLIDE 79B
There were 1¢, 5¢ and 10¢ coins in a piggy box. Three 1¢ coins were removed, four 5¢ nickels were added, and five 10¢ dimes were removed. The value of the money in the piggy box at the end was one third the value at the start. Find the value of the money at the start.
Solution
Let the value at the start be x
Three 1¢ pennies were removed, three 5¢ nickels were added, and five 10¢ dimes were removed ?
The value of the money in the piggy box at the end was one third the value at the start ?
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