Download Introduction to Algebra File

Algebra
The greatest mathematical
tool of all!!
This is a course in basic introductory
algebra.
Essential Prerequisites:
• Ability to work with directed numbers (positives
and negatives)
• An understanding of order of operations
Stephen is 5 years older than Nancy.
Their ages add to 80.
How old are they?
Without algebra, students would probably guess different
pairs of values (with each pair differing by 5) and hope to
somehow find a pair that add to 80. Eventually, we might
work out that Stephen is 42 ½ and Nancy is 37 ½ .
The problem, however, is that there can be too
much guesswork. Algebra takes away the
guesswork.
SEE THE SOLUTION!
Contents
1. Substituting – numerals and
pronumerals
GO!!
2. Like and unlike terms – adding
and subtracting
GO!!
3. Multiplying
GO!!
4. Dividing
GO!!
5. Mixed operations and order of
operations
GO!!
Section 1
Substituting – numerals and pronumerals
When working with algebra, you will meet TWO different
kinds of terms…..
• NUMERALS
These are all the ordinary numbers you’ve
been working with all your life. Numerals
include 2, 5, 7, 235, 15½, 9¾, 2.757, 3.07, –
9, – 7.6 , 0,  and so on.
• PRONUMERALS
These are symbols like , , , and letters
(either single letters or combinations) like x,
y, a, b, ab, xyz, x2 , y3 etc…Pronumerals often
take the place of numerals.
If
a=5
b=2
c=3
find the value of
(1) a + b
(2) c – a – b
SOLUTION
SOLUTION
a+b
=5+2
c–a–b
= 7 (ans)
=–4
=3–5–2
(ans)
NOTE
WHEN WORKING WITH PRONUMERALS YOU’RE
ALLOWED TO LEAVE OUT MULTIPLICATION SIGNS.
PRONUMERALS ARE USUALLY WRITTEN
ALPHABETICALLY (ab rather than ba)
3 x a = 3a
2 x p = 2p
5 x a x b = 5ab
a x b = ab
c x a x b = abc
THIS DOESN’T APPLY WHEN WORKING ONLY
WITH NUMBERS
3 x 4 can’t be written
as 34!
If
a=5
b=2
c=3
find the value of
(3)
ab + c
(4)
4bc – 2a + ab
SOLUTION
SOLUTION
ab + c
=5×2+3
4bc – 2a + ab
=4×2×3–2×5+5×2
= 13 (ans)
= 24 – 10 + 10
= 24
(ans)
Remember to
do
multiplication
first!!
If
a=5
b=2
c=3
find the value of
(5) a(b + c)
(6)
4b(7c – 4a)
SOLUTION
SOLUTION
a (b + c)
= a × (b + c)
4b(7c – 4a)
= 4 × b × (7 × c – 4 × a)
= 5 × (2 + 3)
=5×5
= 4 × 2 × (7 × 3 – 4 × 5)
= 25 (ans)
=8× 1
= 8 × (21 – 20)
= 8
(ans)
REMEMBER
ORDER OF
OPERATIONS
a=–3
b = 10
c=–4
find the value of
If
(1) 3a + b
(2) c – (4a – b)
SOLUTION
SOLUTION
= 3a + b
=3×a+b
= 3 × – 3 + 10
c – (4a – b)
= c – (4 × a – b)
= – 4 – (4 × – 3 – 10)
= – 9 + 10
= – 4 – ( – 12 – 10)
= – 4 – ( – 22)
= – 4 + 22
= 1 (ans)
= 18
(ans)
Note a – (– b) is
same as a + b !!
a=–3
b = 10
c=–4
evaluate
If
MULTIPLY
BEFORE
YOU ADD!!
(4)
(3) a2 + b2
SOLUTION
SOLUTION
= a2 + b2
=a×a+b×b
= – 3 × – 3 + 10 × 10
= 9 + 100
= 109 (ans)
= 2.4
(ans)
Section 2
Like and Unlike terms
Adding and Subtracting
Work out the value of 5 × 7 + 3 × 7.
5 × 7 + 3 × 7 = 35 + 21
= 56
Now work out the value of 8 × 7.
8 × 7 = 56
So here we have two different questions that give the
same answer, 56. So we can make this conclusion:
5×7 + 3×7 = 8×7
5 × 7 + 3× 7 = 8× 7
Or, in words,
5 lots of 7 + 3 lots of 7 = 8 lots of 7
Can you predict the value of 9 × 5 – 2 × 5?
If you said 7 × 5 then you would be correct!
Check that both sums equal 35!
Try these! Make sure you write the SHORT SUM first,
then the answer!
“Long sum” “Short sum”
2x8+5x8
7x8
6x9+2x9
8x9
4x7+1x7
5x7
8x9–3x9
5x9
4x6
7x6–3x6
5x2+2
Rewrite as
5x2+1x2
6x2
Answer
56
72
35
45
24
12
So by now you are hopefully beginning
to see the general pattern. For example
using the fact 3 + 4 = 7 we can write….
3×1 +4×1
3×2 +4×2
3×8 +4×8
3 × 9½ + 4 × 9½
=7×1
=7×2
=7×8
= 7 × 9½
In fact, the pattern holds for all numbers (not just 1, 2,
8 and 9½) and can be written more generally as
3×a +4×a
3×x +4×x
=7×a
=7×x
or
or
any pronumeral (letter) of your choice!
Now try these:
7 × b + 8 × b = 15 × b
2 × y + 9 × y = 11 × y
4×p–2×p= 2 × p
6×q–1×q= 5 × q
8×x+1×x= 9 × x
5 × x
4×x+ x=
8×a–2×a = 6×a
Remember x
really means 1x
What would be a single statement that
would cover all possibilities in this pattern?
8 × 7 + 5 × 7 = 13 × 7
8 × 2 + 5 × 2 = 13 × 2
8 × 12 + 5 × 12 = 13 × 12
And this pattern?
Ans:
8 × a + 5 × a = 13 × a
Of course we could have
used any pronumeral here –
it does not have to be a or w.
5×9– 2×9=3×9
5×2– 2×2=3×2
5 × 79 – 2 × 79 = 3 × 79
Ans: 5 × w – 2 × w = 3 × w
When you’re writing algebra sums, you’re
allowed to LEAVE OUT
MULTIPLICATION SIGNS!
So,
BUT
3 × c can be written as 3c
7 × a × b can be written as 7ab
x × y can be written as xy
3 × 4 CAN’T be written as 34!!
Now try these:
4a + 7a = 11a
5y + 8y = 13y
2p – 6p = – 4p
q + 7q = 8q
x + x = 2x
10x + x = 11x
a – 7a = – 6a
– 7z + 5z = – 2z
Remember x really
means 1x
When terms are multiplied, the
order is not important….
6 × 5 is the same as 5 × 6 (both = 30)
a × b is the same as b × a. i.e. ab = ba. We usually
use alphabetical order though, so ab rather than ba.
a×b×c
=b×a×c
=a×c×b
= c × a × b etc ….
Again, we prefer
alphabetical order
so abc is best.
6 × a is the same as a × 6 i.e. 6a = a6
But the number is usually written first!
Although it’s still correct, we don’t write a6.
Always write 6a. So…. PUT THE NUMBERS
BEFORE PRONUMERALS
More about like terms…
We know that ab and ba are the same thing,
so we can do sums like
4ab + 5ba = 9ab
7xy + 9yx = 16xy
3abc – 2bca + 8cab = 9abc
mnp – 2mpn – 7pmn = – 8mnp
Note that in each case,
• the number goes first
• alphabetical order is used for the answer
(though it is still correct to write 9ba, 16yx etc…)
We know we can write
Key
+ 4 × 5 as a “short sum” 11 × 5,
Question: 7but×is5there
a similar way of writing
6×3+5×7?
If we calculate this sum, it is equal to 18 + 35 = 53
But there are no factors of 53 (other than 1 and 53)
so there is NO SHORT SUM for 6 × 3 + 5 × 7 !!
Because of this, we can conclude that there is no
easy way of writing 6a + 7b, other than 6a + 7b!
In summary, we can simplify 6a + 7a to get 13a.
But cannot simplify 6a + 7b.
6a + 7a is an example of LIKE TERMS.
Like terms can be added or subtracted to
get a simpler answer (13a in this case)
6a + 7b is an example of UNLIKE TERMS.
Like terms cannot be added or subtracted.
You will encounter terms with powers such as x2 ,
3a2 , 5p3, 3a2 b etc. These are treated the same
way as terms with single pronumerals. x2 and x
are UNLIKE, just as x and y are.
2a and 3a2 are UNLIKE and can’t be added or subtracted
3b and 3b4 are UNLIKE and can’t be added or subtracted
2ab and 4ab2 are UNLIKE and can’t be added or subtracted
2a2 and 3a2 are LIKE and can be added to get 5a2. Subtracted to get –1a2 or – a2
9a6 and 4a6 are LIKE and can be added to get 13a6. Subtracted to get 5a6
2a2band 4ba2 are LIKE and can be added to get 6a2b. Subtracted to get – 2a2b
2a2band 4ab2 are UNLIKE and can’t be added or subtracted. They’re unlike
because the powers are on different pronumerals
ab and ac are UNLIKE and can’t be added or subtracted
3a and 5 are UNLIKE and can’t be added or subtracted
3a and 3 are UNLIKE and can’t be added or subtracted
In the table below, match each term from Column 1 with
its “like” term from Column 2
4a
– 11pk2
6w
2kp2
6
7a
ab
9bca
3y
h
7pk2
2c2
– 7d
12c
5c
7d
3h
– 5w
3p2k
5ba
– 3c2
y
4abc
2
Answers next slide
In the table below, match each term from Column 1 with
its “like” term from Column 2
4a
– 11k2p
6w
2kp2
6
7a
ab
9bca
3y
h
7pk2
2c2
– 7d
12c
5c
7d
3h
– 5w
3p2k
5ba
– 3c2
y
4abc
2
Like terms – very important in addition
and subtraction algebra sums !
3a + 2a = 5a
6ab – 2ab = 4ab
7a2 – 3a2 = 4a2
2ac – 7ca = – 5ac
8xy2 – 3xy2 = 5xy2
x – 7x + 2x = – 4x
8x2 – 3x2 = 5x2
8x – 3y = 8x – 3y
5x2 – 3x = 5x2 – 3x
2ab – 3ac = 2ab – 3ac
These questions have
algebra parts that are
like (the same). When
that happens, you can
simplify them!
These questions have
algebra parts that are
different. When that
happens, you can’t
simplify them!
A mixed bag. Which have like terms? Simplify those that do.
5a – 3a
2x + 7x
3x + 8y
4a – 2b
5a + 4
2x + 7x2
3xy + 8yx
4ab – 2b
5a + 4a2
3x – 9x
–xy + 7yx
4ab – 7ba
2a
9x
11xy
– 6x
6xy
–3ab
5a + 6
x + 6x
8x + 8y
a – 7a
5abc + 4cba
2x3 + 2x2
xy + yx
ab – ba
5 + 4a2
x – 9x
4x3 + 5x3
7ac – ca
7x
– 6a
9abc
2xy
0
– 8x
9x3
6ac
Simplifying expressions with more than two terms
3x + 5x + y + 8y
= 8x + 9y (ans)
The like terms are added
together
2a – 3a + 5b – 6b
= – a – b (ans)
The like terms are simplified
5a2 + 3a + 2a2 + a
= 7a2 + 4a (ans)
The like terms are simplified
Remember that terms with a
and a2 are unlike and can’t be
added
Simplify 5 – 7 + 6 – 2
Working left to right
using “cut ‘n’ paste”
5–7+6–2
=–2+6–2
=4–2
=2
But we can also rearrange the terms in the
original question using “cut ‘n’ paste”
First, draw lines to separate the
terms, placing lines in front of
each + or – sign
5
Now, “cut” any term between the
lines (with its sign) and move it to a
new position. We’ll move the “+6”
next to the “5”, swapping it with the
“– 7”
5
–7 +6
–2
–2
Now the original question appears as
5+6–7–2
Which can now easily be simplified to the
correct answer, 2.
Note that we did not HAVE to cut and paste the
+6 and the –7 . We are allowed to cut and
paste ANY TERMS we like.
We will now apply this to help us simplify ALGEBRAIC
EXPRESSIONS, and aim to cut and paste so like terms
are together.
Example 1: Simplify 3a + 2b + 5a – 9b
Here we’ll use cut ‘n’ paste
to bring the a’s together
and the b’s together by
swapping the +2b and +5a
3a
+2b +5a
3a
The question now becomes
Simplifying like terms, we get
3a + 5a = 8a
2b – 9b = – 7b
–9b
–9b
3a + 5a + 2b – 9b
= 8a – 7b ans
Example 2: Simplify a – 9b – 2b + 8a
Again use cut ‘n’ paste to
bring the a’s together and
the b’s together by
swapping the – 9b and +8a
The question now becomes
Simplifying like terms, we get
a + 8a = 9a
– 2b – 9b = – 11b
a
– 9b – 2b +8a
a
–2b
a + 8a – 2b – 9b
= 9a – 11b ans
Example 3: Simplify 2x – 5 + 4x + 8
Again use cut ‘n’ paste to
bring the x’s together and
the numbers together by
swapping the – 5 and +4x
2x – 5
+4x
2x
+8
+8
The question now becomes
2x + 4x – 5 + 8
Simplifying like terms, we get
2x + 4x = 6x
– 5 + 8 = +3
= 6x + 3 ans
Example 4: Simplify 3y – 2x – 5x2 + 4y + x – 2x2
Here there are 6 terms which
3y
can be grouped into 3 pairs of
like terms (2 terms contain an x,
2 terms contain an x2 and 2
terms contain y.
– 2x – 5x2 +4y +x
Swap +4y with – 2x
This puts the y’s together
= 3y
+4y – 5x2 – 2x +x
Now swap +x with – 5x2
This puts the x’s together
and the x2 terms together
= 3y
+4y
Simplifying like terms,
we get
3y + 4y = 7y
x– 2x = – x
– 5x2 – 2x2 = – 7 x2
+x
– 2x2
– 2x2
– 2x – 5x2 – 2x2
= 7y – x – 7x2 ans
Section 3
Multiplying and working with brackets
Remember the basic rules







Place numbers before letters
Keep letters in alphabetical order
Two negatives multiply to make a positive
A negative and a positive multiply to make a negative
If an even number of negatives is multiplied, the
answer is a positive (because they pair off)
If an odd number of negatives is multiplied, the
answer is a negative (one is left after they pair off)
You can rearrange terms that are all being multiplied
(3 x 4 x 5 = 5 x 3 x 4 = 4 x 5 x 3; ab = ba etc…)
When doing multiplication, we do not have to bother
with like terms!
4 × a = 4a
– x × –3y = 3xy
a × 5 × 2 = 10a
– 5x × –3y × – 6p = –90pxy
c × a × b = abc
w × – 3 × k = – 3kw
x × a × b × w = abwx
a × 3bc = 3abc
– 2a × 3b × c × – 5d
– 3b × a = – 3ab
= 30abcd
a × 5c = 5ac
½ a × 5b × 6c = 15abc
2a × b = 2ab
NOTE in this last question, it’s
easier to change the order and do
½a x 6c x 5b
–x×x×x
= – xxx
a×a
= aa
b×b×b
= bbb
= a2
= b3
a2 × a
= aa × a
= aaa
b2 × b × b3 (– x)4
= bbbbbb = (– x)(–x)(– x)(–x)
= b6
= x4
= a3
= – x3
(3a)2
(7b)4
= 3a × 3a
7b × 7b × 7b × 7b
=
9a2
(–5x)3
= 2401b4
(– ab)4
= – 5x × – 5x × – 5x = –ab × –ab × –ab × –ab
= – 125x3
= +a4b4
NOTE: negatives raised to an even power give a POSITIVE
negatives raised to an odd power give a NEGATIVE
QUESTION
ANSWER
QUESTION
2a × b =
–2a × –5a × a =
–a × b × ab × 2ba =
2a × – 3 =
2ab
15ab
–6a
a×–a=
–a2
a2 × a3 =
2 × –a =
–2a
6a2
8p2
–y × –y × –y =
5a × 3b =
3a × 2a =
–4p × –2p =
2a × 4a × 7a =
ab × 3ab =
a × 3ab × 5b × 2 =
ab × –2ab × –3a =
–a × – 3 × –2a × 6b
–4a × 3a2 =
2ab × –3a2b3 =
–cd × –2cd =
ANSWER
10a3
–2a3b3
–12a3
a5
–y3
–6a3b4
2c2d2
2a × 3a × 5a =
30a3
3a2b2
30a2b2
2a + 3a + 5a =
11a
8a3b3
6a3b2
–36a2b
(–5abc)2 =
56a3
(2ab)3 =
(–2cdg)3 =
Note the blue one! It’s an addition!!
25a2b2c2
–8c3d3g3
Section 4
Dividing
Basically, all expressions with a division sign can
be simplified, or at least rewritten in a more
concise form.
Consider the expression 24 ÷ 18.
This can be written as a fraction 24 and simplified
18
further by dividing (cancelling) numerator and
denominator by 6….
4 24
4

318
3
The same process can be applied to algebraic
expressions….
Example 1 Simplify 12x ÷ 3
Solution
3
12x ÷ 3
12x

3
412x

1 3
4x

1
Writing as a fraction
Now think…. What is the largest number that divides
into both numerator and denominator? (the HCF )
= 4x ans
Note when there is only a
“1” left in the
denominator, ignore it!
Example 2 Simplify 8ab ÷ 2a
Solution
8ab ÷ 2a
8ab

2a
4 8ab

1 2a
Writing as a fraction
Dividing numerator and denominator by
HCF 2 and also by a.
= 4b ans
Example 3
Simplify 28abc ÷ 18acd
Solution
28abc ÷ 18acd
28abc

18acd

14 28abc
Writing as a fraction
Dividing numerator and denominator by
HCF (2) and by a and by c.
9 18acd
14b

9d
Note – in this question (and
many others) your answer will
be a fraction!
Example 4
Simplify 20a3b2c ÷ 8ab2c4
Solution
20a3b2c ÷ 8ab2c4
20aaabbc

8abbcccc
5 20aaabbc

2 8abbcccc
5aa

2ccc
2
5a
 3
2c
Writing as a fraction and in “expanded”
format to make dividing easier
Dividing numerator and denominator by
HCF (4) and cancelling matching pairs of
pronumerals (a with a, b with b etc)
Example 5
Simplify 5xy2z ÷ –15x2y3z5
Solution
5xy2z ÷ –15x2y3z5
5 xyyz
Writing as a fraction and in “expanded”
 15 xxyyyzzzzz format to make dividing easier
Dividing numerator and denominator by
5 xyyz
1

HCF (5) and cancelling matching pairs of
– 3  15 xxyyyzzzzz
pronumerals (x with x, y with y etc)

1

 3xyzzzz
1

3 xyz 4
IMPORTANT NOTES
(1) When a negative sign remains in top or
bottom, place it in front of the whole fraction
(2) When only a “1” remains in the top, you
must keep it. (Remember when a “1”
remains in the bottom, you can ignore it)
Section 5
Mixed Operations
B
DM
AS
• Before anything, simplify all BRACKETS
Then..
• Working from left to right, do all
DIVISON and MULTIPLICATION
operations
Then..
• Working from left to right, do
all ADDITION and
SUBTRACTION operations
Example 1
Simplify 20 – 2 × 9
Solution
No brackets.
Do the multiplication first
Now do the subtraction
20 – 2 × 9
= 20 – 18
= 2 (ans)
NOTE – good setting out has all the “=“ signs
directly under one another, and never more
than one “=“ sign on the same line.
Example 2
Simplify 13 – (6 + 5) × 4
Solution
Do brackets first.
Now do the multiplication
Now do the subtraction
13 – (6 + 5) × 4
= 13 –
11
= 13
–
×4
44
= – 31 (ans)
Example 3
Simplify 2x × 3 + 4 × 5x
Solution
No brackets.
Do the two multiplications
working left to right
2x × 3 + 4 × 5x
2x × 3 + 4 × 5x
= 6x + 20x
Now do the addition
Remembering
that you can
only add LIKE
TERMS
= 26x (ans)
Example 4
Simplify 2x × 6xy – 4y × 3x2
Solution
No brackets.
Do the two multiplications
working left to right
Now do the subtraction
2x × 6xy – 4y × 3x2
2x × 6xy – 4y × 3x2
= 12x2y – 12x2y
= 0 (ans)
Example 5
Simplify 12ab – (2b + 3b) × 4a
Solution
12ab – (2b + 3b) × 4a
Do brackets first.
= 12ab –
Now do the multiplication
= 12ab
Now do the subtraction
= – 8ab (ans)
5b × 4a
–
20ab
Example 6
Simplify 8a – 12ab ÷ (4b + 2b) + 3a × a – 5a2
Solution
8a – 12ab ÷ (4b + 2b) + 3a × a – 5a2
Brackets first.
= 8a – 12ab ÷ 6b +
Division &
multiplication
= 8a –
2a
Now subtract like terms
= 6a – 2a2
(ans)
+
3a × a – 5a2
3a2 – 5a2
Example 7
Simplify (3ab2)2 – (ab + ab) × 4ab3 + 2a3b ÷ – ab
Solution
Note (3ab2)2 = 3ab2 x 3ab2 = 9a2b4
(3ab2)2 – (ab + ab) × 4ab3 + 2a3b5 ÷ – ab
Brackets
first
Multiplication
& Division
= 9a2b4
–
= 9a2b4
–
Note 2ab x 4ab3 = 8a2b4
Subtract as
these are all
like terms
= –a2b4
× 4ab3 + 2a3b5 ÷ – ab
2ab
8a2b4
–
2a2b4
Note 2a3b5 ÷ – ab = – 2a2b4
The solution to our introductory problem on
Slide #3
Let Nancy’s age = x
So Stephen’s age = x + 5
These add to 80, so
x + 5 + x = 80
2x + 5 = 80
2x = 75
x = 75 ÷ 2
x = 37 ½
So Nancy’s age is 37½ and
Stephen (who is 5 years older)
must be 42 ½
NO GUESSWORK!!!
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