Algebra Expressions Year 9

Note 1: Expressions

We often use x to represent some number in an equation. We refer to the letter x as a variable.

x + 5 means ‘a number with 5 added on’ x – 7 means ‘a number with 7 subtracted from it’

e.g.

We don’t use the x or ÷ signs in algebra instead we write it like this: 5 x y = 5y x ÷ 9 =

Terms should be written in alphabetical order. xyz and 3b x 4a = 12ab

Note 1: Expressions

A few Rules: A number should be written before a letter. y x 2 = 2y

e.g.

e.g.

e.g. 9x

Three plus a number

Activity: Expressions Match each algebraic expression with the phrase

9x

Five times a number

Half of a number

A number multiplied by seven

A number plus six

A number divided by nine

A number minus eight

Three times a number plus one

2x

5x 3x + 1 7x

3 + x x – 8

x + 6

2x

9x

One plus a number

Activity: Expressions Write an equivalent algebraic expression for each phrase

Twelve times a number

A quarter of a number

A number multiplied by three

A number plus one hundred

A number divided by nineteen

A number minus four

Eight times a number minus one

12x

8x – 1

3x 1 + x

x – 4

x + 100 4x

19x

IWB Ex 11.01 Pg 275-276

Note 2: Substitution

• We replace the variable (letter) with a number and calculate the answer.

• Algebra follows the same rules as BEDMAS!

If a = 2, b = -3, c = 5 then calculate:

a + 5 3b a + b + c = 2 + 5 = 7

= 3 x -3 = -9

= 2 + -3 + 5

= 2 – 3 + 5 = 4

e.g.

Note 2: Substitution

Remember: Do multiplication and division before addition and subtraction Anything in brackets is worked out first A number in front of the bracket means multiply The fraction line means divide

4(d + e) 5def – 2e = 4(3 + 7) = 4 x 10

= 5 x 3 x 7 x -2 – 2 x 7 = -210 – 14

=

= 4

e.g. If d = 3, e = 7, f = -2 then calculate:

= 40 = -224

414 f−

4214 −−

4214 + =

Note 2: Substitution

Remember: Do multiplication and division before addition and subtraction Anything in brackets is worked out first A number in front of the bracket means multiply The fraction line means divide

2(d - e)2 5f – 2d = 2(3 - 7)2 = 2 x (-4)2

= 5 x -2 – 2 x 3

= -10 – 6

=

= 2

e.g. If d = 3, e = 7, f = -2 then calculate:

= -16

773 −e

7773 −×

7721− =

= 2 x 16 = 32

Note 2: Substitution e.g.

Number of tyres = 5x

the number of cars

b Number of tyres = 5 x 60 = 300

c 5 x 40 = 200

Starter

= 5 x 3 = 15

= 2 x -4 = -8

= 6 x 2 = 12

= 5 x -4 = -20

= 12 x 3 = 36

= 3 x 2 = 6

= 3 x -4 = -12

= 2 x -4 = -8

= 3 x 2 x -4 = -24

-

Note 3: Formulas

e.g.

Charge = $10 + $5h

5 hours

A formula is a mathematical rule that explains how to calculate some quantity.

John baby sits for his neighbours. He charges a set fee of $10 plus $5 for every hour (h), that he baby sits. A formula to calculate this charge is given by:

Use the formula to calculate the amount John charges if he baby sits for:

3 hours = 10 + 5 x 5 = $ 35

= 10 + 5 x 3 = $ 25

5 5

Note 3: Formulas

e.g. If John receives $65 how long did he baby sit for?

$65 = $10 + $5 x h

Charge = $10 + $5h

65 = 10 + 5h

65-10 = 10-10 + 5h 55 = 5h

11 = h

John baby sat for 11 hours

IWB Ex 12.01 Pg 299-302 Ex 12.02 Pg 305-308

Starter

Multiply the base length by the height and divide by 2

285× = 20 cm2

Note 4: Multiplying Algebraic Expressions

Rules: Multiply the numbers in the expression (these are written first) Write letters in alphabetical order

Write in simplest form:

y + y + y + y

a + a + a + a + a

a + a + a + a + a + b

a + a + a + a + a + b + b + b

x + y + x + y + x + y + x

x + y + y + y + y - x

= 4y

= 5a

= 5a + b

= 5a + 3b

= 4x + 3y

= 4y

Note 5: Adding & Subtracting Algebraic Expressions

Write in simplest form:

y + 3y

3x + 5x

5x – 2x + 3x

12p + 3r – 2p + 3r

10x + x + 19x

5x – 4x

= 4y

= 8x

= 6x

= 10p + 6r

= 30x

= x

Note 5: Adding & Subtracting Algebraic Expressions

ALPHA Ex 11.06 Pg 161

Note 6: Like and Unlike Terms

Like terms are terms that contain the exact same variables (letters) or combinations of letters.

e.g. Like Terms

2x, 5x, 25x, -81x, 13x, 0.5x…

xy, 2xy, -4xy, ½xy, -100xy,…

2abc, 4bac, 6 cab, 9abc, …..

Remember: If we had written these terms properly (in alphabetical order), it would be more obvious that they are like terms.

Note 6: Like and Unlike Terms

Like terms are terms that contain the exact same variables (letters) or combinations of letters.

e.g. Unlike Terms

2, 2x, 3y

3a, 7ab, 8b

2p, 4r, 10s

Note 6: Like and Unlike Terms

Expressions can have a mixture of LIKE and UNLIKE terms.

e.g. 3x + 5y

Like terms can be grouped together

2a – 7b + 5a + 8b

and simplified

= 7a + b

Unlike terms cannot be simplified

e.g. 5a + 2b – 3a + 6c + 4b 2a + 6b + 6c

Note 6: Like and Unlike Terms

2x + 10y

8x + 8y

4x + 1

6p + 9q

x + 1

2x + 2

x

2x – 7

alpha Ex 11.07 Pg 162 Ex 11.08 Pg 164

14 14

Starter

x

Find the perimeter of this shape in terms of x

The perimeter is the sum of all side lengths

P = x + (2x + 1) + (2x + 3) + (4x + 1) + (5x – 2)

= 14x + 3

If the perimeter is 31 cm. What is the value of x and which side is the longest?

P = 14x + 3 31 = 14x + 3

28 = 14x x = 2

Note 7: Powers (exponents)

Recall: When a variable (letter) is multiplied by itself many times, we use powers

e.g. Write the following in index form:

p x p x p = ________________ q x q x q x q x q = __________ s x s x s x t x t x t x t = _______ p x p x q x q x s x s = ________ s x t x t x t x s x s = __________

p3

q5

s3t4

p2q2s2

s3t3

Note 7: Powers (exponents)

Substituting: Evaluate the following when a =2, b = 7 and c = -3

a2

(b + c)2

5a2

(5a)2

2a2c2

= 22

= (7 - 3)2

= 4

= (4)2 = 16

= (5 x 2)2 = 102 = 100

= 5 x 22 = 5 x 4 = 20

= 2 x 22 x (-3)2

= 2 x 4 x 9

= 72

Note 7: Powers (exponents)

Multiplying:

When multiplying power expressions with the same base, we add the powers.

= (b x b x b) x (b x b x b x b)

= b7

Simplify: b3 x b4

e.g. e2 x e6 g8 x g 5m3 x 4m3 = e2+6

= e8

= g8+1

= g9

= 20m3+3

= 20m6

Note 7: Powers (exponents) Dividing:

When dividing power expressions with the same base, we subtract the powers.

= Simplify:

e.g. g7 ÷ g

= f 6-3

= f 3

= g7-1

= g6 = 5q7-3

= 5q4

2

5

cc

ccccccc

×××××

3

6

ff

3

7

525

qq

2

8

46

ss

= 23 6s

= c3

IWB Ex 13.08 Pg 346 Ex 13.09 Pg 349 PUZZLE Pg 350

Starter How would you calculate 7 x 83 in your head?

7 x 80 + 7 x 3 560 + 21 = 581

What we have done in our head can also look like this:

7 (80 + 3)

x x

= 7 x 80 = 581

+ 7 x 3

Note 8: Expanding Brackets To expand (remove) brackets:

– Multiply the outside term by everything inside the brackets

– Simplify where possible e.g. Expand: a.) 4(x + y) b.) −2(x – y) c.) 5(x – y + 2z)

= 4x + 4y

= -2x + 2y

= 5x - 5y + 10z

The Distributive

Law

Try These! e.g. Expand:

a.) 8(x + y) b.) 4(x – y) c.) 2(x – y) d.) 3(-x + y) e.) 9(x + y + z)

= 8x + 8y

f.) -8(x + y) g.) 5(x – 3y) h.) -(x – 2y) i.) -7(-x + 7y) j.) -4(3x - y + 5z)

= 4x - 4y

= 2x - 2y

= -3x + 3y

= 9x + 9y + 9z

= -8x – 8y

= 5x – 15y

= -x + 2y

= 7x – 49y

= -12x + 4y – 20z

Lets do some more e.g. Expand:

a.) 8(x + 4) b.) 4(x – 2y) c.) 2(x – 10) d.) 3(2x + 9) e.) -(x + 2y - z)

= 8x + 32

f.) -8(x + 8) g.) 5(5x – 3) h.) -(x – 11) i.) -7(x + 12) j.) -2(x - y + 14)

= 4x – 8y

= 2x - 20

= 6x + 27

= -x – 2y + z

= -8x – 64

= 25x – 15

= -x + 11

= -7x – 84

= -2x + 2y – 28

Starter e.g. Expand:

a.) 2(x + y) b.) 4(x – y) c.) 2(x – 3y) d.) 3(x + 30) e.) -(2x - 4y + z)

= 2x + 2y

f.) -8(x + 2) g.) -5(3x – 6) h.) -(x – 23) i.) -3(-2x + 3) j.) -5(x - 2y + 1)

= 4x – 4y

= 2x – 6y

= 3x + 90

= -2x + 4y – z

= -8x – 16

= -15x + 30

= -x + 23

= 6x – 9

= -5x + 10y – 5

Note 8: Expanding Brackets The terms on the inside can also be multiplied by a

variable on the outside.

e.g. Expand: a.) a(x + y) b.) 2a(a + b) c.) 5x2(x2 – x + 2)

= ax + ay

= 2a2 + 2ab

= 5x4 – 5x3 + 10x2

Your turn! e.g. Expand:

a.) a(x + 4) b.) b(x – 5y) c.) x(x – 15) d.) y(2x + 2) e.) x(x2 + 2y – 8)

= ax + 4a

f.) x5(x4+ x3y) g.) 5xy(3xy – 1) h.) -x(x – 11)

= bx – 5by

= x2 – 15x

= 2xy + 2y

= x3 – 2xy – 8x

= x9 + x8y

= 15x2y2 – 5xy

= -x2 + 11x

Note 8: Expanding Brackets and Collecting Like Terms

Expand the brackets first, then simplify!

e.g. Expand & Simplify a.) 4(2x + y) + 3(x + 5y)

= 8x + 4y + 3x + 15y * Collect like terms

= 11x + 19y

b.) 4(5x - y) – 3(x – 10) = 20x - 4y – 3x + 30 = 17x – 4y + 30

* Collect like terms

Your turn!

= 5x + 5y + 2x + 2y = 7x + 7y

= 2x + 2y + 8x + 4y = 10x + 6y

= 6x + 3y + 6x + 12y = 12x + 15y

= 12x + 18y – 10x – 4y = 2x + 14y

= 2x + 6 + 4x + 24 = 6x + 30

IWB - odd only Ex 15.02, 15.03 Pg 389 Ex 15.04 Pg 390 Ex 15.05 Pg 391 Ex 15.06, 15.07 Pg 393 Ex 15.08 Pg 397

Factorising Factorising is the reverse procedure of expanding.

3 (x + 2)

3x + 6

Expanding

Factorising

Note 9: Factorising (put in brackets)

Factorising is the reverse process of expanding. • We want to put brackets back into the algebraic expression • find the highest common factor and write it in front of

the brackets

e.g. Factorise

3x + 3y 4x – 4y 7x + 7y + 7z = 3( ) x+y = 4( ) x – y = 7( ) x + y + z

e.g. Factorise: a.) 6a + 6b b.) 3p – 3q c.) 4x + 4y

= 6( )

= 3( )

= 4( )

d.) 6x + 12

e.) 24x + 24y = 24( )

= 6( )

Try These

a+b

p – q

x+y

x+2

x+y

f.) 7x +7 g.) 7x + 14 h.) 24x + 36

= 7( ) x+1

= 7( ) x+2

= 12( ) 2x+3

You can check that your answer is correct by expanding

e.g. Factorise: a.) 8a + 6b b.) 12p – 3q c.) 4x + 8

= 2( )

= 3( )

= 4( )

d.) 6x + 30

e.) 29x + 29 = 29( )

= 6( )

Try These

4a+3b

4p – q

x+2

x+5

x+1

f.) 7x + 49 g.) 9x + 63 h.) 45x + 81

= 7( ) x+7

= 9( ) x+7

= 9( ) 5x+9

You can check that your answer is correct by expanding

Factorise: a.) 4a + 8b b.) 3p – 6q + 3r c.) 4x + 8y + 12z

= 4(a+2b)

= 3(p – 2q +r)

= 4(x + 2y + 3z)

d.) 6x + 21

e.) 24x - 32 = 8(3x – 4)

= 3(2x +7)

Starter

IWB - odd only Ex 15.11 Pg 400 Ex 15.13 Pg 401 Ex 15.14 Pg 402 Ex 15.15 Pg 403 Ex 15.16 Pg 405

How do we simplify an exponential term raised to another exponent?

= (2y3) × (2y3)

= 4y6

= (3a4) × (3a4) × (3a4)

= 27a12

Extension – More exponent rules

e.g. (2y3)2

e.g. (3a4)3

× ×

Notice that there is a shortcut to get the same result

= 22y2×3

= 4y6

= 33a4×3

= 27a12

1.) Index the number. 2.) Multiply each variable index by the index

outside the brackets. 3.) If the bracket can be simplified, do this first. e.g. Simplify

(2x2)3 = 23 x2×3 = 8x6

(-4h2g6)2 = (-4)2h2×2g6×2

22

312

xx

= (4x)2

= 16x2

Extension – More exponent rules

= 16h4g12

× ×

QUADRATIC EXPANSION When we expand two brackets we use: F – first (multiply the first variable or number from each bracket) O – outside (multiply the outside variables together) I – inside (multiply the two inside variables together) L – last (multiply the last variable in each bracket together) Simplify, leaving your answer with the highest power first to the

lowest power (or number) last.

e.g. (x + 4) (x – 2)

F O I L

= x2 - 2x + 4x - 8

= x2 + 2x - 8

Extension – Expanding 2 Brackets

QUADRATIC EXPANSION e.g. (x + 3) (x – 5)

= x2 + 3x - 5x - 15 = x2 - 2x - 15

e.g. (x + 10) (x + 1)

= x2 + 10x + x + 10 = x2 + 11x + 10

e.g. (x - 3) (x – 8)

= x2 - 3x - 8x + 24 = x2 - 11x + 24

e.g. (x - 4) (x + 4)

= x2 - 4x + 4x - 16 = x2 - 16

•Notice the middle term cancels out DIFFERENCE OF SQUARES

Extension – Expanding 2 Brackets

QUADRATIC EXPANSION e.g. (x + 7) (x – 9)

= x2 + 7x - 9x - 63 = x2 - 2x - 63

e.g. (x – 5) (x + 4)

= x2 - 5x + 4x - 20 = x2 - x - 20

e.g. (x - 2) (x – 6)

= x2 - 2x - 6x + 12 = x2 - 8x + 12

e.g. (x - 9) (x + 9)

= x2 - 9x + 9x - 81 = x2 - 81 •Notice the middle term cancels out

DIFFERENCE OF SQUARES

Extension – Expanding 2 Brackets