Download tpc maths (part a) - nswtmth307a

TPC MATHS (PART A) - NSWTMTH307
TOPIC 5: ALGEBRAIC EXPRESSIONS
AND EQUATIONS
5.1 Algebraic Expressions
A variable is a symbol used to represent a number in an expression or an equation (e.g.
x). The value of this number can change. An algebraic expression is a mathematical
expression that consists of variables, numbers and operations (e.g. 3x+2). The value of
this expression can change.
We can make algebraic expressions from word problems:
Note that 3x means
3 times x
Example 1: Write each phrase as an algebraic expression.
Phrase
Expression
nine increased by a number x
9+x
fourteen decreased by a number p 14 - p
seven less than a number t
t-7
the product of 9 and a number n
9 · n or 9n
thirty-two divided by a number y
32 ÷ y or
Or some harder ones:
Example 2: Write each phrase as an algebraic expression using the variable n.
Phrase
Expression
five more than twice a number
2n + 5
the product of a number and 6
6n
seven divided by twice a number
7 ÷ 2n or
three times a number decreased by 11 3n - 11
Here are some commonly used words and the mathematical operation they refer to:
Operation
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Words Used
+
Addition, Add, Sum, Plus, Increase, Total
-
Subtraction, Subtract, Minus, Less, Difference, Decrease, Take
Away, Deduct
×
Multiplication, Multiply, Product, By, Times, Lots Of
÷
Division, Divide, Quotient, Goes Into, How Many Times
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5.2 Operations in Algebra
Adding and Subtracting:
In algebra, only LIKE TERMS can be added or subtracted.
What do we mean when we say "TERMS"? A term is any part of an algebraic expression
which is joined by a plus or minus sign to another part of the expression.
For example, the expression:
6x + 4 has two terms
5x + 4y + z3 has three terms, etc
And what do we mean when we say "LIKE TERMS"? Like terms are terms whose variables,
and their exponents (indices) are the same.
For example, the following are pairs of like terms:
5x and -3x
6m2 and 0.75m2
Remember: Only
LIKE TERMS can be
added or subtracted
3 p4
p and
4
4
But the following are not like terms:
2x and x2
m2 and m3
p and 4
So, to add or subtract like terms, just pretend the variable is a fruit and you know that one
apple plus one apple = two apples and you can't add apples and oranges to get
appleoranges, so use the same principles with your variables:
x + x = 2x
The + or - sign to the left of a
6m2 - 2m2 = 4m2
4
4
4
term belongs to that term!
8 - 5p + 2 - p = 10 - 6p
Use your number line to work
out addition and subtraction.
Multiplying:
Some basics:
m x m = m2
m x m x m = m3
5 x m = 5m
To multiply algebraic terms (they don't have to be like terms):
- decide whether the answer is a plus or a minus
- multiply the numbers
- multiply each of the variables
For Example:
6m x 7m = 42m2
-3mn x -5m = 15m2n
4m2 x -8m = -32m3
Remember:
+x+=+
-x- =+
+x-=-
Dividing:
Some basics:
m  m=1
m  2=
m
2
or 2  m =
2
m
m2  m = m
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To divide algebraic terms (they don't have to be like terms):
- decide whether the answer is a plus or a minus
- divide the numbers
- divide each of the variables
For Example:
6m  2 = 3m
-30mn  -5m = 6n
40m2 x -8m = -5m
Remember:
+  +=+
-  - =+
+  -=-
Fractions:
Rule: To multiply fractions, cancel out, then multiply the numerators and multiply the
denominators
Example:
7 m 4n 1  2


 2 Can you see how we cancelled out to get this?
2n 7 m 1  1
Rule: To divide fractions, invert (turn upside down) the second fraction, then multiply.
8m2 4m 8m2 n
2 m  1 2m
Example:





3n
n
3n 4m
3 1
3
5.3 Substitution
In algebra, substituting means putting numbers in place of the variables (letters).
Example: Substitute 3 for m in the expression 4m - 2
4 x 3 - 2 = 10
Example: Substitute 6 for b and 8 for h in the formula for the area of a triangle A =
A=
1
bh
2
1
x 6 x 8 = 24, i.e. the area of the triangle is 24
2
Be wary of substituting negative numbers
e.g. if m = -5, then the expression 3m2 becomes
3 x 25 = 75
5.4 Expanding
In algebra, expanding means multiplying to get rid of the brackets.
The expression 3(m+2) means 3 multiplied by each term inside the brackets,
so, 3(m+2) = 3 x m + 3 x 2
= 3m + 6
If you see
Some
1.
2.
3.
more examples of expanding:
5(n-3) = 5n - 15
-p(p+4) = -p2 - 4p
6(s+3t-4)+5s = 6s + 18t - 24 + 5s = 11s + 18t - 24
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-(m-4), it is the
same as:
-1(m-4) = -m+4
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5.5 Factorising
In algebra, factorising is the opposite of expanding - it means taking a factor outside the
brackets.
For example, if we have the expression 5m + 10, we can see that a common factor for the
two terms is 5, so we take this outside the brackets and we get 5(m+2)
Some more examples:
1. -4n - 8 = -4(n+2)
2. -8pq+12p2 = -4p(2q-3p)
5.6 Solving Linear Equations
An equation is an expression which contains an equal sign and it says that whatever is on
the left of the equals sign is the same value as whatever is on the right.
Equations usually contain variables and we can solve the equation to find the value of the
variable.
The Ultimate Rule for Solving
Equations:
Whatever you do to one side
of the equation, you MUST do
to the other.
For example, if we have m + 2 = 5, we can see that m=3. This is the solution to the
equation.
What we really did was to subtract 2 from both sides of the equation:
m+2-2
=5-2
m
=3
The Steps:
1. Aim to get the variable by itself.
2. To do this, do the opposite operation to whatever is with the variable and do it to
both sides of the equation.
3. Repeat until you have the variable by itself on one side of the equation.
Examples: Solve
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m - 8 = 12
m
= 12 + 8
m
= 20
(+8 to both sides)
(simplify)
Opposite Operations
Solve 5 + t = -8
t
= -8 - 5
t
= -13
(-5 from both sides)
(simplify)
Solve 7y
y
m
(divide both sides by 7)
(simplify)
= 140
= 140  7
= 20
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+ and x and 
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Solve
n
5
=9
(multiply both sides by 5)
n
n
=9x5
= 45
(simplify)
Sometimes, you need two (or more) steps to solve the equation.
Example:
Solve
3h + 6 = 21
3h
= 15
h
=5
(-6 from both sides)
(divide both sides by 3)
5.7 Transposing Formulae
If you can solve equations, then you can transpose formulae.
Transposing just means changing the subject of the formula which is whatever variable is
on the left of the equals sign by itself.
To transpose, just follow the steps above as if you were solving the equation.
Example:
Transpose the following formula so that 's' becomes the subject:
m = 4s + n
(subtract n from both sides)
m - n = 4s
(divide both sides by 4)
mn
=s
4
(swap sides)
s
=
mn
4
5.8 Problems involving Substitution and Solving Equations
Often, algebra is used to solve word problems. Here are some hints of what you should do:
1. Make a note of all the important information in the question
2. Work out what the question is asking you
3. Write an algebraic expression or equation to represent the problem
4. Solve for the variable you are after
Example:
A brother is twice as old as his sister. Their combined ages are 24. How old
is the sister?
Solution:
Let the age of the sister be represented by 's'.
Then the age of the brother is 2s.
Write an equation:
s + 2s = 24
And solve it:
3s
= 24
s
=8
The sister's age is 8 years old.
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