Introduction to Algebra

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Introduction to Algebra
On completion of this module you will be able to

• understand what algebra is
• identify and define a variable in a simple word
  problem
• perform some basic mathematical operations with
  variables
• work with algebraic fractions
• understand exponents and radicals involving
  variables
• do some simple factoring

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What is algebra?

• Algebra
• is a branch of maths where symbols are used to
  represent unknown quantities
• is a symbolic language designed to solve
  problems more easily
• involves manipulating expressions involving
  symbols and numbers
• usually uses letters, called variables, to
  represent unknown quantities
• uses , -, ?, ? etc (just as last week)

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Example
Imagine Allan needs to save 1000 to buy a new
stereo. He wants to purchase it in six weeks
time and is going to save an equal amount of
money each week. Lets name the amount he saves
each week using a variable, x. Then he will save
x each week, so
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If we write this using the variable then Be
careful that your x looks different from your
?. The ? is implied if we write This is an
algebraic equation that summarises the word
problem.
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Next we want to determine the value of the
variable that makes this expression
true. Divide both sides of the equations by 6
cancels out the 6 in front of the x.
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• So Allan needs to save 166.67 each week for the
  next six weeks in order to have enough money for
  the stereo.
• QMA this week will focus on identifying
  variables and manipulating algebraic expressions.
• In Week 3 we will look more at solving equations
  using algebra.

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Identifying the variable

• The first step in solving a word problem is to
  identify the unknown (variable).
• Next choose a letter to represent this variable
  often x is used, but many times we use
  something more meaningful t for time, p for
  price, q for quantity, r for (interest) rate
  etc.
• Well look at a couple of examples of
  identifying a variable.

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Example
Rhonda owns a small health food shop. If she
hires another employee to serve behind the
counter, it will cost her 18 per hour in
wages. Based on her current sales revenue she can
afford to pay an extra 250 per week in
wages. For how many hours each week can she
afford to employ an extra person?
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Solution
The clue to finding the variable is in the
question of the last sentence how many
hours. There are many ways we could express our
variable but one example is Let x be the number
of hours per week that Rhonda can afford to
employ an extra person.
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Example
Matthew received 45 on his first science test
for the year. Because he was concerned about that
result, he studied hard and received 65 and 82
respectively on the next two tests. There is one
final test for the year. Matthews final grade
will be awarded based on the average mark for the
four tests. If he wants to receive at least 65
overall, what is the minimum mark he must receive
on the final test to reach his goal?
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Solution
The clue is again in the last sentence what is
the minimum mark he must receive on the final
tests to reach his goal?. Let m be the minimum
mark Matthew must receive on the final test in
order to attain 65 overall.
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Example
John rides his bike from his house to work each
day. If he travels an average speed of 32km per
hour and it takes him 20 minutes to get to work,
how far does he travel? Solution Let d be the
distance from Johns house to his work.
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Example

• Write an algebraic expression to represent the
  following ideas
• Twice a number.
• A number minus five.
• Half a number plus one.
• The area of a rectangle is the length times the
  width.
• Solutions
• 2n

d)
c)
b) n-5
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Example

• Write a phrase to describe the following
  expressions
• Solutions
• A number plus two.
• A quarter of a number or a number divided by 4.
• Four times a number minus five.

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Evaluation

• Evaluation (or substitution) means evaluating an
  expression at a certain value of a variable.
• Example
• Evaluate A lw when l 10 and w 5.
• Substitute a 3, b 2 and c 6 into

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Algebraic Operations

• With algebra we can add or subtract like
  terms.
• For example, 3t and 2t are like terms
  because they both contain t s.
• We add like terms by adding the coefficients.
• So with 3t 2t, the 3 and 2 are coefficients and
• Similarly

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Example

The 6 is not a like term
The a and b terms are not like terms
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Examples
Collect like terms and simplify
Simplify
First group like terms together so it is easier
to add and subtract
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Example
Simplify
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• Recall the rules for multiplying and dividing
  negative numbers (see Week 1).
• The same rules apply to variables.

Example
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• Expanding Brackets
• The same rules apply to variables as did with
  numbers.

Examples
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Examples
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Examples
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The following identities are important
Expand the left hand sides to show how the left
hand sides are equal to the right hand sides
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Fractions
Adding and subtracting fractions

• To add or subtract fractions we convert to
  equivalent fractions with the same denominator
  (as with number fractions).

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Example
Multiplying the denominators together gives
(or use LCD28)
Since both numerator and denominator divide by 4.
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Example
The common denominator is found by multiplying
denominators together
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Multiplying and dividing fractions

• To multiply fractions multiply numerators
  together and multiply denominators together (as
  in Week 1).

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• To divide fractions, multiply by the reciprocal.

Since the x s cancel.
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Exponents and Radicals

• As before, means
• Any variable raised to the power of zero always
  equals one, so

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Multiplying Powers
A more general form of the rules from Week 1 is
given here. If you are
1.
multiplying bases with the same exponent, then
multiply the bases and raise them to this
exponent.
2.
multiplying the same base with different
exponents, then add the exponents.
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3.
Raising a number to an exponent and then to
another exponent, then multiply the exponents.
Another useful rule for dividing numbers with the
same base but different exponents subtract the
exponents
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Taking the root of a number (fractional
exponents)
The square root of a number is the reverse of
squaring.
Another way of writing roots is using
fractional powers
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Negative exponents
A power with a negative exponent can be rewritten
as one over the same number with a positive
exponent
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Examples of negative fractional exponents
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Example
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Example (alternate solution)
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Example
Simplify and express your solution in terms of
positive exponents
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(No Transcript)
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Alternate solution
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Factoring

• Look for common factor in two or more
  expressions
• E.g. in ab ac, the common factor is a.
• If we divide each term by a we get
• and .
• We can therefore rewrite ab ac as a(b c).

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Examples
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Putting it all together
Examples

• Simplify
• Simplify
• Simplify and express as a single fraction

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Solutions

1. Simplify

1. Simplify

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1. Simplify and express as a single fraction

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Examples

• Expand the brackets and collect like terms
• Simplify

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Solutions

1. Expand the brackets and collect like terms

1. Simplify

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Factoring Trinomials (Optional)

• This section will help with solving quadratic
  equations in later weeks, but is not essential
  (you will not lose any marks for not using this
  technique).
• If you are familiar with factoring trinomials
  (from school) then it would be wise to work
  through this section.
• If factoring trinomials is new to you and you
  are interested, then work through this section.