Download Basic Algebra Skills

Basic Algebra Skills
Numeracy Workshop
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Introduction
These slides are intended to give you a basic introduction to algebra.
Drop-in Study Sessions: Monday, Wednesday, Thursday, 10am-12pm, Meeting
Room 1.15, First floor, Guild Building, every week.
Ask a Maths Question: See the website.
Website: Slides, worksheet, solutions, online quiz.
www.studysmarter.uwa.edu.au → Numeracy → Online Resources
Email: [email protected]
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Introduction
In the Number Skills Workshop, we said that Addition is commutative. This
means that when you add two numbers together, the order does not matter.
3 + 17 = 17 + 3
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Introduction
In the Number Skills Workshop, we said that Addition is commutative. This
means that when you add two numbers together, the order does not matter.
3 + 17 = 17 + 3
Statements such as Addition is commutative, are mathematical in nature, and so
we wish to express these statements in the form of expressions and equations,
rather than English.
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Introduction
Instead of saying Addition is commutative, we can say that for any two numbers x
and y , we have that
x +y =y +x
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Introduction
Instead of saying Addition is commutative, we can say that for any two numbers x
and y , we have that
x +y =y +x
Using x and y to represent numbers, means we don’t have to be specific about
which numbers we use. We can write our rules using pronumerals rather than
numerals.
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Multiplication
xy means “x times y ”.
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Multiplication
xy means “x times y ”.
5x means “5 times x” or “5 lots of x”.
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Multiplication
xy means “x times y ”.
5x means “5 times x” or “5 lots of x”.
5xy 2 means “5 times x times y squared”.
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Multiplication
xy means “x times y ”.
5x means “5 times x” or “5 lots of x”.
5xy 2 means “5 times x times y squared”.
4
y
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means “4 divided by y ”.
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Multiplication
xy means “x times y ”.
5x means “5 times x” or “5 lots of x”.
5xy 2 means “5 times x times y squared”.
4
y
2x
y
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means “4 divided by y ”.
means “2 times x, divided by y ”.
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Variables
In the previous slides, x and y are called variables.
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Variables
In the previous slides, x and y are called variables.
This is because we haven’t identified them with specific numbers, and so they
represent a whole range of different numbers. In other words, they vary.
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Variables
In the previous slides, x and y are called variables.
This is because we haven’t identified them with specific numbers, and so they
represent a whole range of different numbers. In other words, they vary.
Variables are place-holders for numbers
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Expressions
An expression is a combination of numbers and variables.
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Expressions
An expression is a combination of numbers and variables.
Some examples are:
5x
4
3x + 5
2xy + 4x 2 y + 2x + 4x
√
7x 2 − x + x1
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Terms
Expressions are made up of terms.
4x 2 + 2xy −
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4x
+ 7xy − 2x 2
y
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Terms
Expressions are made up of terms.
4x 2 + 2xy −
4x
+ 7xy − 2x 2
y
The terms are the pieces of the above equation which are being added or
subtracted. We will put a box around each term:
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Terms
Every term in the expression is connected to the sign which is directly to
the left of it.
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Terms
Every term in the expression is connected to the sign which is directly to
the left of it.
Here is an expression:
4x 2 +2xy −
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4x
+7xy −2x 2
y
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Terms
Every term in the expression is connected to the sign which is directly to
the left of it.
Here is an expression:
4x 2 +2xy −
4x
+7xy −2x 2
y
We see that 2xy is connected to the + sign, and
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4x
y
is connected to the − sign.
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Terms
Every term in the expression is connected to the sign which is directly to
the left of it.
Here is an expression:
4x 2 +2xy −
4x
+7xy −2x 2
y
We see that 2xy is connected to the + sign, and
4x
y
is connected to the − sign.
We see that 4x 2 has no sign to the left of it, but this absence of a sign means
that it is positive, and so in a sense there is a hidden + sign.
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Terms
We put a box around each term so that each term is coupled with its sign.
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Terms
We put a box around each term so that each term is coupled with its sign.
We can move these blocks around to change how the expression looks, while
keeping it equivalent.
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Terms
We put a box around each term so that each term is coupled with its sign.
We can move these blocks around to change how the expression looks, while
keeping it equivalent.
If we move the block containing 4x 2 from the left, we need to give it a + sign.
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Moving terms
We can change how an expression looks by moving the terms around. When we
move a term, we must keep it connected to the same sign, that is, the sign on the
left of the term stays with it wherever it goes.
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Moving terms
We can change how an expression looks by moving the terms around. When we
move a term, we must keep it connected to the same sign, that is, the sign on the
left of the term stays with it wherever it goes.
These three expressions are equivalent:
4x 2 + 2xy −
4x 2 −
4x
+ 7xy − 2x 2
y
4x
− 2x 2 + 7xy + 2xy
y
2xy − 2x 2 + 7xy + 4x 2 −
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y
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Exercise
Out of the following four expressions, which two are the same?
2x 2 + 4x − 7 − 2xy
−2xy + 4x − 7 − 2x 2
4x − 2x 2 − 7 − 2xy
−7 + 4x + 2xy − 2x 2
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Exercise
Out of the following four expressions, which two are the same?
2x 2 + 4x − 7 − 2xy
−2xy + 4x − 7 − 2x 2
4x − 2x 2 − 7 − 2xy
−7 + 4x + 2xy − 2x 2
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Like Terms
Some terms are made up using the same combination of variables, with
matching variables having matching powers. These are called like terms.
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Like Terms
Some terms are made up using the same combination of variables, with
matching variables having matching powers. These are called like terms.
The terms 4xy and 17xy are like terms, as both consist of the same variables.
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Like Terms
Some terms are made up using the same combination of variables, with
matching variables having matching powers. These are called like terms.
The terms 4xy and 17xy are like terms, as both consist of the same variables.
The terms 4xy 2 and 3xy 2 are like terms. Both consist of the same variables, and
in both cases the y is being squared.
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Like Terms
The terms −6w 3 xyz 2 and 22w 3 xyz 2 are like terms. Both consist of the same
variables, and in both cases the w is being cubed, and the z is being squared.
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Like Terms
The terms −6w 3 xyz 2 and 22w 3 xyz 2 are like terms. Both consist of the same
variables, and in both cases the w is being cubed, and the z is being squared.
The terms 7 and 3 are like terms, as they are both simply numbers.
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Like Terms
The terms −6xy and yx are like terms. Both consist of the same variables. The
order in which they appear is irrelevant. This is because multiplication is
commutative.
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Like Terms
The terms −6xy and yx are like terms. Both consist of the same variables. The
order in which they appear is irrelevant. This is because multiplication is
commutative.
The terms 52y 2 x 3 z and −15zy 2 x 3 are like terms. Both consist of the same
variables. The order in which they appear is irrelevant. It is also important to
note that in both terms, y is being squared, x is being cubed, and z is single.
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Unlike Terms: Examples
The terms 5 and 6x are unlike. One contains an x whereas the other does not.
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Unlike Terms: Examples
The terms 5 and 6x are unlike. One contains an x whereas the other does not.
The terms 4xy 2 and −2xy are unlike. In the first term, y is being squared but in
the second term, y is not.
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Unlike Terms: Examples
The terms 5 and 6x are unlike. One contains an x whereas the other does not.
The terms 4xy 2 and −2xy are unlike. In the first term, y is being squared but in
the second term, y is not.
The terms 2x 2 y and 3y 2 x are unlike. In the first term, x is being squared, but in
the second term y is being squared.
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Adding and Subtracting Like Terms
We can add and subtract like terms!
7xy + 4xy = 11xy
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Adding and Subtracting Like Terms
We can add and subtract like terms!
7xy + 4xy = 11xy
9x 2 y + 5yx 2 = 14x 2 y
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Adding and Subtracting Like Terms
We can add and subtract like terms!
7xy + 4xy = 11xy
9x 2 y + 5yx 2 = 14x 2 y
14xyz 3 − 5z 3 yx + 2yz 3 x = 11z 3 yx
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Adding and Subtracting Like Terms
Note that when we see expressions without a number at the front, such as x, xy 2
and wxz, these really mean 1x, 1xy 2 and 1wxz. We usually just don’t write in
the 1.
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Adding and Subtracting Like Terms
Note that when we see expressions without a number at the front, such as x, xy 2
and wxz, these really mean 1x, 1xy 2 and 1wxz. We usually just don’t write in
the 1.
2xy + xy = 3xy
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Adding and Subtracting Like Terms
Note that when we see expressions without a number at the front, such as x, xy 2
and wxz, these really mean 1x, 1xy 2 and 1wxz. We usually just don’t write in
the 1.
2xy + xy = 3xy
14x 2 y − yx 2 = 13x 2 y
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Adding and Subtracting Like Terms
Note that when we see expressions without a number at the front, such as x, xy 2
and wxz, these really mean 1x, 1xy 2 and 1wxz. We usually just don’t write in
the 1.
2xy + xy = 3xy
14x 2 y − yx 2 = 13x 2 y
xyz + yzx + yxz = 3xyz
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Simplifying Expressions
We can use our knowledge of like terms, and shifting around terms, to simplify
big scary expressions!
Example: Simplify the following expression:
4xy 2 + 3xy + 2xy 2 − 2yx
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Simplifying Expressions
We can use our knowledge of like terms, and shifting around terms, to simplify
big scary expressions!
Example: Simplify the following expression:
4xy 2 + 3xy + 2xy 2 − 2yx
How many distinct terms are there? We see that 4xy 2 and 2xy 2 are like terms,
and also 3xy and −2yx are like terms.
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Simplifying Expressions
We can use our knowledge of like terms, and shifting around terms, to simplify
big scary expressions!
Example: Simplify the following expression:
4xy 2 + 3xy + 2xy 2 − 2yx
How many distinct terms are there? We see that 4xy 2 and 2xy 2 are like terms,
and also 3xy and −2yx are like terms.
Remembering to keep the signs the same, we shift the expression around to put
like terms next to each other:
4xy 2 + 2xy 2 + 3xy − 2yx
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Simplifying Expressions
4xy 2 + 2xy 2 + 3xy − 2yx
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Simplifying Expressions
4xy 2 + 2xy 2 + 3xy − 2yx
Now we can use our knowledge of adding and subtracting like terms to make it
simpler. We know that 4xy 2 + 2xy 2 = 6xy 2 , and also that 3xy − 2yx = xy . Doing
this we get:
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Simplifying Expressions
4xy 2 + 2xy 2 + 3xy − 2yx
Now we can use our knowledge of adding and subtracting like terms to make it
simpler. We know that 4xy 2 + 2xy 2 = 6xy 2 , and also that 3xy − 2yx = xy . Doing
this we get:
6xy 2 + xy
Our expression has been simplified into something which looks slightly less scary.
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Simplifying Expressions: Example
Simplify the following expression:
xy 2 − 3xyz + 2y 2 x + 5yzx − 4yxz
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Simplifying Expressions: Example
Simplify the following expression:
xy 2 − 3xyz + 2y 2 x + 5yzx − 4yxz
Step 1: Identify like terms:
xy 2 − 3xyz + 2y 2 x + 5yzx − 4yxz
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Simplifying Expressions: Example
Simplify the following expression:
xy 2 − 3xyz + 2y 2 x + 5yzx − 4yxz
Step 1: Identify like terms:
xy 2 − 3xyz + 2y 2 x + 5yzx − 4yxz
Step 2: Move terms so that like terms are next to each other:
xy 2 + 2y 2 x − 3xyz + 5yzx − 4yxz
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Simplifying Expressions: Example
xy 2 + 2y 2 x − 3xyz + 5yzx − 4yxz
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Simplifying Expressions: Example
xy 2 + 2y 2 x − 3xyz + 5yzx − 4yxz
Step 3: Now add the like terms together:
3xy 2 − 2yxz
We have simplified our original expression!
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Multiplying Terms
We have seen that if terms are like, then we can add or subtract them.
We would also like to multiply terms together.
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Multiplying Terms
We have seen that if terms are like, then we can add or subtract them.
We would also like to multiply terms together.
It turns out, that two terms do not have to be like to be multiplied together.
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Multiplying Terms: Example
Calculate the following multiplication of terms: (3x 2 y 3 )(4x 5 y 2 ).
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Multiplying Terms: Example
Calculate the following multiplication of terms: (3x 2 y 3 )(4x 5 y 2 ).
We multiply the numbers together, we multiply the x’s together, and multiply the
y ’s together.
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Multiplying Terms: Example
Calculate the following multiplication of terms: (3x 2 y 3 )(4x 5 y 2 ).
We multiply the numbers together, we multiply the x’s together, and multiply the
y ’s together.
Numbers are easy, we know that 3 × 4 = 12.
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Multiplying Terms: Example
Calculate the following multiplication of terms: (3x 2 y 3 )(4x 5 y 2 ).
We multiply the numbers together, we multiply the x’s together, and multiply the
y ’s together.
Numbers are easy, we know that 3 × 4 = 12.
Now we need to multiply x 2 by x 5 . Remember that x 2 really means xx (x times
x) and x 5 really means xxxxx. Multiplying these we get:
x 2 x 5 = xxxxxxx = x 7
We just add the powers together!
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Multiplying Terms: Example
Calculate the following multiplication of terms: (3x 2 y 3 )(4x 5 y 2 ).
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Multiplying Terms: Example
Calculate the following multiplication of terms: (3x 2 y 3 )(4x 5 y 2 ).
Now to multiply the y ’s together. We now know that we just add the powers, so
we have:
y 3y 2 = y 5
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Multiplying Terms: Example
Calculate the following multiplication of terms: (3x 2 y 3 )(4x 5 y 2 ).
Now to multiply the y ’s together. We now know that we just add the powers, so
we have:
y 3y 2 = y 5
Putting this all together we now know how to multiply the terms together:
(3x 2 y 3 )(4x 5 y 2 ) = 12x 7 y 5
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Multiplying Terms: Examples
(2x 2 y )(3xy 4 ) = 6x 3 y 5
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Multiplying Terms: Examples
(2x 2 y )(3xy 4 ) = 6x 3 y 5
(−x 3 y 3 )(xy 5 ) = −x 4 y 8
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Multiplying Terms: Examples
(2x 2 y )(3xy 4 ) = 6x 3 y 5
(−x 3 y 3 )(xy 5 ) = −x 4 y 8
(−5xy )(−2x 2 y 3 ) = 10x 3 y 4
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Multiplying Terms: Examples
(2x 2 y )(3xy 4 ) = 6x 3 y 5
(−x 3 y 3 )(xy 5 ) = −x 4 y 8
(−5xy )(−2x 2 y 3 ) = 10x 3 y 4
(−2x 2 y )(−3x 4 y )(−4y ) = −24x 6 y 3
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Single Bracket Expansion
We have just seen how to simplify expressions which look like:
term × term
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Single Bracket Expansion
We have just seen how to simplify expressions which look like:
term × term
Sometimes, we have to work with expressions of the form
term ×( term + term )
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Single Bracket Expansion
We have just seen how to simplify expressions which look like:
term × term
Sometimes, we have to work with expressions of the form
term ×( term + term )
With numbers we use BIMDAS, and do what’s in the brackets first. However, the
two terms in the brackets can only be added together if they are like terms.
If the terms are unlike, then we need to cheat BIMDAS using expansion.
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Single Bracket Expansion: Example
Expand the following: 4(3 + x)
This looks like term ×( term + term ), where two of the terms are just numbers.
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Single Bracket Expansion: Example
Expand the following: 4(3 + x)
This looks like term ×( term + term ), where two of the terms are just numbers.
We will use the distributive law, which tells us how to get rid of the brackets,
without being able to add 3 and x with each other, as they are unlike.
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Single Bracket Expansion: Example
Expand the following: 4(3 + x)
This looks like term ×( term + term ), where two of the terms are just numbers.
We will use the distributive law, which tells us how to get rid of the brackets,
without being able to add 3 and x with each other, as they are unlike.
Distributive Law: a(b+c) = ab + ac
where a, b and c all represent terms.
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Single Bracket Expansion: Example
Expand the following: 4(3 + x)
Basically, each term inside the brackets gets multiplied by the term out the
front. As all of our terms are positive, multiplying them will keep them positive.
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Single Bracket Expansion: Example
Expand the following: 4(3 + x)
Basically, each term inside the brackets gets multiplied by the term out the
front. As all of our terms are positive, multiplying them will keep them positive.
So we will multiply 4 by 3 to get 12, and we will multiply 4 by x to get 4x.
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Single Bracket Expansion: Example
Expand the following: 4(3 + x)
Basically, each term inside the brackets gets multiplied by the term out the
front. As all of our terms are positive, multiplying them will keep them positive.
So we will multiply 4 by 3 to get 12, and we will multiply 4 by x to get 4x.
Expanding the brackets gives:
4(3 + x) = 12 + 4x
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Single Bracket Expansion: Examples
4(2x + 3) = 8x + 12
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Single Bracket Expansion: Examples
4(2x + 3) = 8x + 12
3(4 − 2x) = 12 − 6x
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Single Bracket Expansion: Examples
4(2x + 3) = 8x + 12
3(4 − 2x) = 12 − 6x
4x(7 + 5x) = 28x + 20x 2
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Single Bracket Expansion: More examples
−2xy (3x − 2y ) = −6x 2 y + 4xy 2
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Single Bracket Expansion: More examples
−2xy (3x − 2y ) = −6x 2 y + 4xy 2
−4x 2 y 3 (6xy + 3x 3 y 3 ) = −24x 3 y 4 − 12x 5 y 6
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Single Bracket Expansion: More examples
We can even expand expressions of the form:
term ×( term + term + term )
Once again, we multiply each term in the bracket by the term out the front.
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Single Bracket Expansion: More examples
2x(3 + x + 5x 4 ) = 6x + 2x 2 + 10x 5
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Single Bracket Expansion: More examples
2x(3 + x + 5x 4 ) = 6x + 2x 2 + 10x 5
−4xy (2xy − 4x + 3x) = −8x 2 y 2 + 16x 2 y − 12x 2 y
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Double Bracket Expansion
Now we look at double bracket expansion, which involves expanding expressions
of the form:
( term + term ) × ( term + term )
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Double Bracket Expansion
Now we look at double bracket expansion, which involves expanding expressions
of the form:
( term + term ) × ( term + term )
Each term in the first set of brackets gets coupled up with each term in the
second set of brackets.
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Double Bracket Expansion
Now we look at double bracket expansion, which involves expanding expressions
of the form:
( term + term ) × ( term + term )
Each term in the first set of brackets gets coupled up with each term in the
second set of brackets.
There are two terms to choose from in each, so there are 4 possible
combinations in total.
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Double Bracket Expansion: Example
Expand the following:
(3 + 2x)(5 + 3x)
We need to calculate each of the following:
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Double Bracket Expansion: Example
Expand the following:
(3 + 2x)(5 + 3x)
We need to calculate each of the following:
3 times 5
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Double Bracket Expansion: Example
Expand the following:
(3 + 2x)(5 + 3x)
We need to calculate each of the following:
3 times 5
3 times 3x
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Double Bracket Expansion: Example
Expand the following:
(3 + 2x)(5 + 3x)
We need to calculate each of the following:
3 times 5
3 times 3x
2x times 5
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Double Bracket Expansion: Example
Expand the following:
(3 + 2x)(5 + 3x)
We need to calculate each of the following:
3 times 5
3 times 3x
2x times 5
2x times 3x
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Double Bracket Expansion: Example
We can do each of these:
3 times 5 equals 15
3 times 3x equals 9x
2x times 5 equals 10x
2x times 3x equals 6x 2
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Double Bracket Expansion: Example
We can do each of these:
3 times 5 equals 15
3 times 3x equals 9x
2x times 5 equals 10x
2x times 3x equals 6x 2
So we can expand as follows:
(3 + 2x)(5 + 3x) = 15 + 9x + 10x + 6x 2
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Double Bracket Expansion: Example
We can do each of these:
3 times 5 equals 15
3 times 3x equals 9x
2x times 5 equals 10x
2x times 3x equals 6x 2
So we can expand as follows:
(3 + 2x)(5 + 3x) = 15 + 9x + 10x + 6x 2
Usually, once we expand we simplify like terms, so we get:
15 + 19x + 6x 2
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Double Bracket Expansion: Example
Expand the following:
(4x − 5)(2 + 4x)
We need to calculate each of the following:
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Double Bracket Expansion: Example
Expand the following:
(4x − 5)(2 + 4x)
We need to calculate each of the following:
4x times 2
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Double Bracket Expansion: Example
Expand the following:
(4x − 5)(2 + 4x)
We need to calculate each of the following:
4x times 2
4x times 4x
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Double Bracket Expansion: Example
Expand the following:
(4x − 5)(2 + 4x)
We need to calculate each of the following:
4x times 2
4x times 4x
−5 times 2
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Double Bracket Expansion: Example
Expand the following:
(4x − 5)(2 + 4x)
We need to calculate each of the following:
4x times 2
4x times 4x
−5 times 2
−5 times 4x
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Double Bracket Expansion: Example
We can do each of these:
4x times 2 equals 8x
4x times 4x equals 16x 2
−5 times 2 equals − 10
−5 times 4x equals − 20x
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Double Bracket Expansion: Example
We can do each of these:
4x times 2 equals 8x
4x times 4x equals 16x 2
−5 times 2 equals − 10
−5 times 4x equals − 20x
So we can expand as follows:
(4x − 5)(2 + 4x) = 8x + 16x 2 − 10 − 20x
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Double Bracket Expansion: Example
We can do each of these:
4x times 2 equals 8x
4x times 4x equals 16x 2
−5 times 2 equals − 10
−5 times 4x equals − 20x
So we can expand as follows:
(4x − 5)(2 + 4x) = 8x + 16x 2 − 10 − 20x
Usually, once we expand we simplify like terms, so we get:
16x 2 − 12x − 10
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Double Bracket Expansion: Examples
(3 − 4x)(2x − 7) = 6x − 21 − 8x 2 + 28x
= −8x 2 + 34x − 21
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Double Bracket Expansion: Examples
(3 − 4x)(2x − 7) = 6x − 21 − 8x 2 + 28x
= −8x 2 + 34x − 21
(6 − y )(4 + x) = 24 + 6x − 4y − xy
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Double Bracket Expansion: Examples
(3 − 4x)(2x − 7) = 6x − 21 − 8x 2 + 28x
= −8x 2 + 34x − 21
(6 − y )(4 + x) = 24 + 6x − 4y − xy
(6x + 7xy )(3x 2 − 5y 3 ) = 18x 3 − 30xy 3 + 21x 3 y − 35xy 4
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Equations
An equation is a mathematical object of the form:
expression = expression
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Equations
An equation is a mathematical object of the form:
expression = expression
For example:
3x 2 y + 6xy = 7xy 2 − 4y
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Equations
An equation which involves the variable x, tells us how x engages with other
numbers.
From this, we might be able to deduce the identity of x. This is called solving the
equation for x.
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Equations
An equation which involves the variable x, tells us how x engages with other
numbers.
From this, we might be able to deduce the identity of x. This is called solving the
equation for x.
Perhaps we are given the following equation:
x +5=8
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Equations
An equation which involves the variable x, tells us how x engages with other
numbers.
From this, we might be able to deduce the identity of x. This is called solving the
equation for x.
Perhaps we are given the following equation:
x +5=8
This equation tells us that when you add 5 to x, you end up with 8. There is only
one known number which acts like this and that number is 3. So x must be equal
to 3. We write this as:
x =3
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Equations
What if we were given this equation:
4 + 2x = 20
This says, when you add four with two times x, you end up with 20.
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Equations
What if we were given this equation:
4 + 2x = 20
This says, when you add four with two times x, you end up with 20.
We know from experience, that adding four with sixteen gives us 20.
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Equations
What if we were given this equation:
4 + 2x = 20
This says, when you add four with two times x, you end up with 20.
We know from experience, that adding four with sixteen gives us 20.
So this 2x has been put there in place of 16. We now know that two times x is
sixteen, and so x must be equal to 8. We write:
x =8
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Equations
Sometimes equations can be solved as we did above, by thinking.
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Equations
Sometimes equations can be solved as we did above, by thinking.
Other times they are slightly harder to work through.
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Equations
Sometimes equations can be solved as we did above, by thinking.
Other times they are slightly harder to work through.
We wish to have a process which will allow us to systematically solve linear
equations, perhaps the most common type.
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Linear Equations
Linear equations are equations which only contain number terms like 3 and 6 and
single variables with no power such as 2x and −4y .
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Linear Equations
Linear equations are equations which only contain number terms like 3 and 6 and
single variables with no power such as 2x and −4y .
Here are some examples:
4 + 3x = 20
3 − 5y = −2
17w = 3 − w
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Solving Linear equations
An equation is a statement that two quantities are the same. For example:
2x + 3 = 15
The above equation says that 2x + 3 is equal to 15, and our task is to find the
value of x that makes this so.
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Solving Linear equations
An equation is a statement that two quantities are the same. For example:
2x + 3 = 15
The above equation says that 2x + 3 is equal to 15, and our task is to find the
value of x that makes this so.
As both sides of the equation represent the exact same quantity, adding or
subtracting the same number to both sides will keep the equation true. We may
also multiply or divide both sides by the same number.
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Solving Linear Equations
This idea of tweaking both sides by the same amount proves most useful, and is
the basis for solving all sorts of equations in algebra.
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Solving Linear Equations
This idea of tweaking both sides by the same amount proves most useful, and is
the basis for solving all sorts of equations in algebra.
2x + 3 = 15
We are trying to change the above equation to a new equation, a simpler one of
the form
x = number.
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Solving Linear Equations
This idea of tweaking both sides by the same amount proves most useful, and is
the basis for solving all sorts of equations in algebra.
2x + 3 = 15
We are trying to change the above equation to a new equation, a simpler one of
the form
x = number.
The way we get to this new equation is to move all the numbers away from x,
so that it is sitting on a side of the equals sign all by itself.
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Solving Linear Equations: Example
Solve the following equation for x:
2x + 3 = 15
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Solving Linear Equations: Example
Solve the following equation for x:
2x + 3 = 15
The first thing to do is to move away the numbers which are not in the same
term as x.
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Solving Linear Equations: Example
Solve the following equation for x:
2x + 3 = 15
The first thing to do is to move away the numbers which are not in the same
term as x.
The 3 is being added to it is a different term.
The 2 is part of the same term as x.
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Solving Linear Equations: Example
Solve the following equation for x:
2x + 3 = 15
The first thing to do is to move away the numbers which are not in the same
term as x.
The 3 is being added to it is a different term.
The 2 is part of the same term as x.
In this sense, 2 and x are closer to each other (they are part of the same term),
and so we should move the 3 away first.
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Solving Linear Equations: Example
2x + 3 = 15
We basically want to subtract away 3 from the left hand side of the equation.
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Solving Linear Equations: Example
2x + 3 = 15
We basically want to subtract away 3 from the left hand side of the equation.
But we know, that we must do the same operation to both sides, to preserve the
equation.
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Solving Linear Equations: Example
2x + 3 = 15
We basically want to subtract away 3 from the left hand side of the equation.
But we know, that we must do the same operation to both sides, to preserve the
equation.
So we take away 3 from both sides:
2x + 3−3 = 15−3
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Solving Linear Equations: Example
2x + 3 = 15
We basically want to subtract away 3 from the left hand side of the equation.
But we know, that we must do the same operation to both sides, to preserve the
equation.
So we take away 3 from both sides:
2x + 3−3 = 15−3
Which gives us
2x = 12
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Solving Linear Equations: Example
2x = 12
It is probably pretty clear now that the answer is 6. However, we still need to
complete our method, so that we can complete harder problems later on.
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Solving Linear Equations: Example
2x = 12
It is probably pretty clear now that the answer is 6. However, we still need to
complete our method, so that we can complete harder problems later on.
We now need to get rid of the 2. We divide both sides by 2 here to get:
12
2x
=
2
2
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Solving Linear Equations: Example
2x
12
=
2
2
If you look at the left hand side of the above equation, we can see that x is being
multiplied by 2, and then divided by 2. But doubling a number and then halving
it, gets you back to where you started. So the left hand side is really just x. So
our equation is:
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Solving Linear Equations: Example
2x
12
=
2
2
If you look at the left hand side of the above equation, we can see that x is being
multiplied by 2, and then divided by 2. But doubling a number and then halving
it, gets you back to where you started. So the left hand side is really just x. So
our equation is:
x=
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12
2
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Solving Linear Equations: Example
2x
12
=
2
2
If you look at the left hand side of the above equation, we can see that x is being
multiplied by 2, and then divided by 2. But doubling a number and then halving
it, gets you back to where you started. So the left hand side is really just x. So
our equation is:
x=
12
2
We know that 12 divided by 2 is 6. So we end up with:
x =6
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Solving Linear Equations: Example
Solve the following equation for x:
10 − 2x = 18
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Solving Linear Equations: Example
Solve the following equation for x:
10 − 2x = 18
First we get rid of the 10 on the LHS, by subtracting 10 from both sides:
10 − 2x−10 = 18−10
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Solving Linear Equations: Example
Solve the following equation for x:
10 − 2x = 18
First we get rid of the 10 on the LHS, by subtracting 10 from both sides:
10 − 2x−10 = 18−10
Which we simplify to get:
−2x = 8
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Solving Linear Equations: Example
−2x = 8
Now we divide both sides by −2:
8
−2x
=
−2
−2
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Solving Linear Equations: Example
−2x = 8
Now we divide both sides by −2:
8
−2x
=
−2
−2
Which becomes:
x=
8
−2
So the solution of our equation is x = −4.
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Using STUDYSmarter Resources
This resource was developed for UWA students by the STUDYSmarter team for
the numeracy program. When using our resources, please retain them in their
original form with both the STUDYSmarter heading and the UWA crest.
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