Algebra

The greatest mathematical tool of all!!

AN INTRODUCTION TO

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

IntroductionStephen 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

2. Like and unlike terms – adding and subtracting

3. Multiplying

4. Dividing

5. Mixed operations and order of operations

GO!!

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GO!!

GO!!

Section 1

Substituting – numerals and pronumerals

When working with algebra, you will meet TWO different kinds of terms…..

• NUMERALSThese 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.

EXAMPLE 1

If a = 5b = 2c = 3

find the value of

(1) a + b (2) c – a – b

a + b

= 5 + 2

= 7 (ans)

c – a – b

= 3 – 5 – 2

= – 4 (ans)

SOLUTION SOLUTION

NOTEWHEN 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

a x b = ab 5 x a x b = 5ab

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 = 5b = 2c = 3

find the value of

(3) ab + c (4) 4bc – 2a + ab

ab + c= 5 × 2 + 3

= 13 (ans)

4bc – 2a + ab = 4 × 2 × 3 – 2 × 5 + 5 × 2

= 24 (ans)

SOLUTION SOLUTION

= 24 – 10 + 10 Remember to do multiplication first!!

If a = 5b = 2c = 3

find the value of

(5) a(b + c) (6) 4b(7c – 4a)

a (b + c)

= 5 × (2 + 3)

= 25 (ans)

4b(7c – 4a)

= 4 × 2 × (7 × 3 – 4 × 5)

= 8 (ans)

SOLUTION SOLUTION

= 8 × (21 – 20)= 5 × 5

= 8 × 1

= 4 × b × (7 × c – 4 × a)= a × (b + c)

EXAMPLE 2If a = – 3

b = 10c = – 4

find the value of

(1) 3a + b (2) c – (4a – b)

= 3 × a + b

= – 9 + 10

= 1 (ans)

c – (4a – b)

= – 4 – (4 × – 3 – 10)

= 18 (ans)

SOLUTION SOLUTION

= 3 × – 3 + 10

= 3a + b

REMEMBERORDER OFOPERATIONS

= c – (4 × a – b)

= – 4 – ( – 12 – 10) = – 4 – ( – 22)

= – 4 + 22 Note a – (– b) is same as a + b !!

If a = – 3b = 10c = – 4

evaluate

(3) a2 + b2

(4)

= a × a + b × b

= 9 + 100

= 109 (ans)

= 2.4 (ans)

SOLUTION

SOLUTION

= – 3 × – 3 + 10 × 10

= a2 + b2

MULTIPLY BEFOREYOU ADD!!

 

 

 

 

 

Section 2

Like and Unlike terms

Adding and Subtracting

Work out the value of 5 × 7 + 3 × 7.

5 × 7 + 3 × 7

= 56Now 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

= 35 + 21

5 × 7 + 3× 7 = 8× 7Or, 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” Answer

2 x 8 + 5 x 8 7 x 8 56

6 x 9 + 2 x 9

4 x 7 + 1 x 7

8 x 9 – 3 x 9

7 x 6 – 3 x 6

5 x 2 + 2

8 x 9

5 x 7

5 x 9

4 x 6

Rewrite as5 x 2 + 1 x 2

6 x 2

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….

THE GENERAL PATTERN

3 × 1 + 4 × 1 = 7 × 1 3 × 2 + 4 × 2 = 7 × 2 3 × 8 + 4 × 8 = 7 × 8 3 × 9½ + 4 × 9½ = 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 = 7 × a or3 × x + 4 × x = 7 × x or

any pronumeral (letter) of your choice!

Now try these:

7 × b + 8 × b =2 × y + 9 × y =4 × p – 2 × p =6 × q – 1 × q =

8 × x + 1 × x =4 × x + x =8 × a – 2 × a =

15 × b11 × y2 × p5 × q

9 × x5 × x

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 Ans:8 × a + 5 × a = 13 × a

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

And this pattern? Of course we could have used any pronumeral here – it does not have to be a or w.

And now for some good news

When you’re writing algebra sums, you’re allowed to LEAVE OUT

MULTIPLICATION SIGNS!

So, 3 × c can be written as 3c

7 × a × b can be written as 7ab

x × y can be written as xy

BUT 3 × 4 CAN’T be written as 34!!

Now try these:

4a + 7a =5y + 8y =2p – 6p =q + 7q =

x + x =10x + x =a – 7a =

11a13y

– 4p8q

2x11x– 6a

Remember x really means 1x

– 7z + 5z = – 2z

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 ….

6 × a is the same as a × 6 i.e. 6a = a6But 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

Again, we prefer alphabetical order so abc is best.

More about like terms…

We know that ab and ba are the same thing, so we can do sums like

4ab + 5ba =

7xy + 9yx =

3abc – 2bca + 8cab =

9ab

16xy9abc

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…)

Like and Unlike TermsKey Question:

We know we can write

7 × 5 + 4 × 5 as a “short sum” 11 × 5, but is there a similar way of writing

6 × 3 + 5 × 7 ? If we calculate this sum, it is equal to 18 + 35 = 53But 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.

Like and Unlike Terms

Like and Unlike TermsYou 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

4a – 11pk2

6w 2kp2

6 7a

ab 9bca

3y h

7pk2 2c2

– 7d 12c

5c 7d

3h – 5w

3p2k 5ba

– 3c2 y

4abc 2

In the table below, match each term from Column 1 with its “like” term from Column 2

Answers next slide

4a – 11k2p

6w 2kp2

6 7a

ab 9bca

3y h

7pk2 2c2

– 7d 12c

5c 7d

3h – 5w

3p2k 5ba

– 3c2 y

4abc 2

In the table below, match each term from Column 1 with its “like” term from Column 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

8x – 3y = 8x – 3y

5x2 – 3x = 5x2 – 3x

2ab – 3ac = 2ab – 3ac

8x2 – 3x2 = 5x2

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 – 3a2x + 7x3x + 8y

4a – 2b

5a + 42x + 7x2

3xy + 8yx

4ab – 2b

5a + 4a2

3x – 9x–xy + 7yx

4ab – 7ba

5a + 6x + 6x8x + 8y

a – 7a

5abc + 4cba2x3 + 2x2

xy + yx

ab – ba

5 + 4a2

x – 9x 4x3 + 5x3

7ac – ca

2a9x

11xy

– 6x 6xy

–3ab

7x

– 6a

9abc

2xy 0

– 8x 9x3

6ac

MORE ADVANCED EXAMPLES

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

REARRANGING TERMSSimplify 5 – 7 + 6 – 2

Working left to right 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”

using “cut ‘n’ paste”

5 –7 –2 +6 First, draw lines to separate the terms, placing lines in front of each + or – sign

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 –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 –9b

3a –9b

The question now becomes 3a + 5a + 2b – 9b

= 8a – 7b ansSimplifying like terms, we get3a + 5a = 8a2b – 9b = – 7b

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

a – 9b – 2b +8a

a –2b

The question now becomes a + 8a – 2b – 9b

= 9a – 11b ansSimplifying like terms, we geta + 8a = 9a– 2b – 9b = – 11b

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 +8

2x +8

The question now becomes 2x + 4x – 5 + 8

= 6x + 3 ansSimplifying like terms, we get2x + 4x = 6x– 5 + 8 = +3

Example 4: Simplify 3y – 2x – 5x2 + 4y + x – 2x2

Here there are 6 terms which can be grouped into 3 pairs of like terms (2 terms contain an x, 2 terms contain an x2 and 2 terms contain y.

= 7y – x – 7x2 ansSimplifying like terms, we get3y + 4y = 7yx– 2x = – x – 5x2 – 2x2 = – 7 x2

3y – 2x – 5x2 +4y +x – 2x2

Now swap +x with – 5x2

= 3y – 2x– 5x2+4y +x – 2x2 Swap +4y with – 2x

= 3y – 2x – 5x2+4y +x – 2x2

This puts the y’s together

This puts the x’s together and the x2 terms together

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…)

All algebraic terms can be multipliedWhen doing multiplication, we do not have to bother

with like terms!

– x × –3y =4a4 × a =

3xy

c × a × b = abca × 5 × 2 =

– 3kw w × – 3 × k =

10a

x × a × b × w = abwxa × 3bc = 3abc– 3b × a = – 3ab

a × 5c = 5ac

2a × b = 2ab

– 5x × –3y × – 6p = –90pxy

– 2a × 3b × c × – 5d = 30abcd

½ a × 5b × 6c = 15abcNOTE in this last question, it’s easier to change the order and do½a x 6c x 5b

When you multiply two or more of the same pronumeral…..

a × a = aa = a2

b × b × b = bbb = b3

– x × x × x = – xxx = – x3

a2 × a = aa × a

= a3

b2 × b × b3 = bbbbbb = b6

(– x)4 = (– x)(–x)(– x)(–x) = x4 = aaa

Working with brackets…..

(3a)2

= 3a × 3a

= 9a2

7b × 7b × 7b × 7b

(7b)4

= 2401b4

(–5x)3

= – 5x × – 5x × – 5x

= – 125x3

(– ab)4

= –ab × –ab × –ab × –ab

= +a4b4

NOTE: negatives raised to an even power give a POSITIVE

negatives raised to an odd power give a NEGATIVE

Mixed examples - multiplying

QUESTION ANSWER QUESTION ANSWER

2a × b = –2a × –5a × a =

5a × 3b = –a × b × ab × 2ba =

2a × – 3 = –4a × 3a2 =

a × – a = a2 × a3 =

2 × –a = –y × –y × –y =

3a × 2a = 2ab × –3a2b3 =

–4p × –2p = –cd × –2cd =

2a × 4a × 7a = 2a × 3a × 5a =

ab × 3ab = 2a + 3a + 5a =

a × 3ab × 5b × 2 = (2ab)3 =

ab × –2ab × –3a = (–5abc)2 =

–a × – 3 × –2a × 6b (–2cdg)3 =

2ab

15ab

–6a

–a2

–2a

6a2

8p2

56a3

3a2b2

30a2b2

6a3b2

–36a2b

10a3

–2a3b3

–12a3

a5

–y3

–6a3b4

2c2d2

30a3

11a8a3b3

25a2b2c2

–8c3d3g3

Note the blue one! It’s an addition!!

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 and simplified

further by dividing (cancelling) numerator and denominator by 6….

18

24

18

24

3

4

4

3

The same process can be applied to algebraic expressions….

Simplify 12x ÷ 3

3

12x

3

12x

Example 1

Solution

12x ÷ 3

Writing as a fraction

= 4x ans

Now think…. What is the largest number that divides into both numerator and denominator? (the HCF )

4

1

Note when there is only a “1” left in the denominator, ignore it!

3

1

4x

Simplify 8ab ÷ 2a

a

ab

2

8

a

ab

2

8

Example 2

Solution8ab ÷ 2a

Writing as a fraction

= 4b ans

Dividing numerator and denominator by HCF 2

4

and also by a.1

acd

abc

18

28

Simplify 28abc ÷ 18acd

acd

abc

18

28

Example 3

Solution28abc ÷ 18acd

Writing as a fraction

Dividing numerator and denominator by HCF (2) and by a and by c.

d

b

9

14

14

9

Note – in this question (and many others) your answer will be a fraction!

abbcccc

aaabbc

8

20

Simplify 20a3b2c ÷ 8ab2c4

abbcccc

aaabbc

8

20

Example 4

Solution20a3b2c ÷ 8ab2c4

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)

ccc

aa

2

5

5

2

3

2

2

5

c

a

xxyyyzzzzz

xyyz

15

5

Simplify 5xy2z ÷ –15x2y3z5

xxyyyzzzzz

xyyz

15

5

Example 5

Solution5xy2z ÷ –15x2y3z5

Writing as a fraction and in “expanded” format to make dividing easier

Dividing numerator and denominator by HCF (5) and cancelling matching pairs of

pronumerals (x with x, y with y etc)

xyzzzz3

1

1

– 3

43

1

xyz

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

When doing more complicated sums with a mixtured of the four operations (and brackets) you must observe the ORDER OF OPERATIONS RULES….

• 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

B

DM

AS

Example 1

Solution

Simplify 20 – 2 × 9

No brackets.

Do the multiplication first 20 – 2 × 9= 20 – 18

Now do the subtraction = 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

Solution

Simplify 13 – (6 + 5) × 4

Do brackets first.

Now do the multiplication

= 13 – 11 × 4

= 13 – 44

Now do the subtraction = – 31 (ans)

13 – (6 + 5) × 4

Example 3

Solution

Simplify 2x × 3 + 4 × 5x

No brackets.

Do the two multiplications working left to right

= 6x + 20x

= 26x (ans)Now do the addition

2x × 3 + 4 × 5x2x × 3 + 4 × 5x

Remembering that you can only add LIKE TERMS

Example 4

Solution

Simplify 2x × 6xy – 4y × 3x2

No brackets.

Do the two multiplications working left to right

= 12x2y – 12x2y

= 0 (ans)Now do the subtraction

2x × 6xy – 4y × 3x2 2x × 6xy – 4y × 3x2

Example 5

Solution

Simplify 12ab – (2b + 3b) × 4a

Do brackets first.

Now do the multiplication

= 12ab – 5b × 4a

= 12ab – 20ab

Now do the subtraction = – 8ab (ans)

12ab – (2b + 3b) × 4a

Example 6

Solution

Simplify 8a – 12ab ÷ (4b + 2b) + 3a × a – 5a2

Brackets first.

Division &multiplication

= 8a – 12ab ÷ 6b + 3a × a – 5a2

= 8a – 2a + 3a2 – 5a2

Now subtract like terms

= 6a – 2a2 (ans)

8a – 12ab ÷ (4b + 2b) + 3a × a – 5a2

Example 7

Solution

Simplify (3ab2)2 – (ab + ab) × 4ab3 + 2a3b ÷ – ab

Brackets first

(3ab2)2 – (ab + ab) × 4ab3 + 2a3b5 ÷ – ab

= 9a2b4 – 2ab × 4ab3 + 2a3b5 ÷ – ab

Note (3ab2)2 = 3ab2 x 3ab2 = 9a2b4

Multiplication & Division

= 9a2b4 – 8a2b4 – 2a2b4

Note 2ab x 4ab3 = 8a2b4 Note 2a3b5 ÷ – ab = – 2a2b4

= –a2b4 Subtract as these are all like terms

The solution to our introductory problem on Slide #3

Let Nancy’s age = x

So Stephen’s age = x + 5These add to 80, so

x + 5 + x = 802x + 5 = 80

2x = 75x = 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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