topic 9: expressions, formulae & equations fileform 2 – expressions, formulae & equations...

Form 2 – Expressions, Formulae & Equations

Topic 9: Expressions, Formulae & Equations Ms Briffa

Algebra is when letters are involved in maths.

A variable represents a number that changes (varies).

9.1 Simplifying Expressions

1. Simplify the following expressions:

(a) 3b + 4b + b

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(b) 4c + c – 2c + 3c (c) 10p – 5 + 2p + 7 – 4p

________________________________________ ________________________________________

(d) 2 + 3y – 2 + 3 + 7 (e) 4p – q + 5r + 2q – r

________________________________________ ________________________________________

2. Find and simplify an expression for the perimeter of this rectangle.

_____________________________________________________________________________

3. (a) Write down and simplify an expression for the sum of these angles:

__________________________________________________________________________

(b) What do the angles add up to? _________________________________________

(c) What is the value of ? ________________________________________________

4. How should you write the following expressions?

(a) p × r = ______________________________

(b) g × 3 × h = ______________________________

(c) 2 × × = ______________________________

(d) d × e × 5 = ______________________________

ALWAYS write

numbers before letters

1 is the same as

To simplify expressions we collect like terms.

e.g. 2a + 4a = 6a but

2a + 7b cannot be made any simpler

5 4

2c + 3

5c - 2

8b

6c

5 + 3y + 5

8p + 2

4p + q + 4r

(2c + 3) + (5c -2) + (2c + 3) + (5c -2) = 14c + 2

+ 4 + 5 = 10

360°

10 = 360° = 36°

pr or rp

2

3gh or 3hg

5de or 5ed



9.2 Expanding Brackets

Expand and simplify the following:

(a) 3( + 3) (b) 5(y – 1)

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(c) 2(3f + 4) (d) 4(3 – 2 )

________________________________________ ________________________________________

(e) – 5(a + 4) (f) – 2 (6 – 2d)

________________________________________ ________________________________________

(g) 5 + 4( + 3) (h) 3(4 – ) – 2(1 + 3 )

________________________________________ ________________________________________

(i) 3a – (b – a) (j) ( + y) – ( – y)

________________________________________ ________________________________________

1. Multiply what is outside the bracket by everything inside the bracket

2. Simplify the expression

e.g. 6(2a + 3) = 6 x (2a + 3)

= (6 x 2a) + (6 x 3)

= 12a + 18

ALWAYS work out the brackets first!

3 + 9 5y – 5

6f + 8 12 - 8

- 5a - 20 - 12 + 4d

5 + 4 + 12

9 + 12

12 – 3 – 2 – 6

10 – 9

3a – b + a

4a - b

+ y – + y

0 + 2y

2y



9.3 Factorising Brackets

Factorise the following fully:

(a) 4 + 12 (b) 6y – 10

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(c) 36 + 18f (d) 24b – 36c

________________________________________ ________________________________________

9.4 Substituting in a Formula

This means that we replace letters by numbers

When a = 7 , b = 4 , x = 2 and y = - 5

Work out the values of:

a + b + y = ________________________________________ a = ____________________________________________

3a – 4 = ___________________________________________ b + 2 = _________________________________________

3 y = _____________________________________________ 3 + y = _________________________________________

2b – = ___________________________________________ 2a + 3y = ______________________________________

ay + b = _________________________________________ 2a y = __________________________________________

Factorising is the reverse of expanding brackets.

First 'take out' any common factors

5 + 5 = 5( + 1)

2 – 8 = 2( – 4)

4(x + 3) 2(3y – 5)

18(2 + f) 12(2b – 3c)

7 + 4 + - 5 = 6

(3 x 7) – 4 = 21 – 4 = 17

3 x 2 x - 5 = - 30

(2 x 4) – 2 = 8 – 2 = 11

(7 x - 5) + (4 x 2) = - 35 + 8 = - 27

7 x 2 = 14

(4 x 2) + 2 = 8 + 2 = 10

(3 x 2) + - 5 = 6 - 5 = 1

(2 x 7 x 2) + (3 x - 5) = 28 + - 15 = 13

2 x 7 x 2 x - 5 = - 140



9.5 Solving Equations

An equation contains an equals sign (=)

It is in fact like a balanced set of scales, where the left side = right side

(a) + 3 = 7 (b) y – 4 = 10 (c) 4a = 8 (d)

= 3

______________________ ______________________ ______________________ _______________________

______________________ ______________________ ______________________ _______________________

Now try to solve these equations:

e) 2 + 5 = 11 f) 8p – 2 = 22

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

g) 7 = 3s - 5 h)

+ 3 = 15

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

i) Kelly is years old. In 9 years time she will be 27. Form an equation to find her age.

__________________________________________________________________________________________________________

To solve equations, we must ALWAYS keep the equation balanced.

Therefore we add/subtract/multiply/divide the same quantity on BOTH SIDES

+ 3 – 3 = 7 – 3 y – 4 + 4 = 10 + 4 4a ÷ 4 = 8 ÷ 4

= 4 y = 14 a = 2 = 6

x 2 = 3 x 2

2 + 5 – 5 = 11 – 5

2 = 6

= 3

8p – 2 + 2 = 22 + 2

8p = 24

p = 3

7 + 5 = 3s – 5 + 5

12 = 3s

4 = s

+ 3 – 3 = 15 – 3

= 12

= 24

+ 9 = 27 + 9 – 9 = 27 – 9 = 18



It doesn’t matter which

side you group the letters

and numbers! 9.6 Equations with an Unknown on Both Sides

To solve an equation we need to arrive to letter = number.

So when we have an unknown on both sides, collect letters to one side and numbers to the other.

a) 5 = 2 + 9 b) 4y – 8 = 2y

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

c) 3p + 18 = 6p - 6 d) 2 – 7a = a - 14

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

e) Amy chooses a number, multiplies it by 2 and adds 5. Ramona chooses the same number, multiplies it by 3

and subtracts 1. If they get the same answer:

Write an equation to describe their work. _______________________________________________________________

Solve the equation to find their chosen number.

__________________________________________________________________________________________________________

__________________________________________________________________________________________________________

__________________________________________________________________________________________________________

9.7 Solving Equations with Brackets

Whenever brackets are involved, multiply the brackets first!

a) 3( + 2) = 12 b) 39 = 3(2c – 5)

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

5 - 2 = 2 + 9 - 2

3 = 9

= 3

3p + 18 – 3p = 6p – 6 – 3p

18 + 6= 3p – 6 + 6

24 = 3p

4y – 8 – 4y = 2y – 4y

8 = 2y

4 = y

2 – 7a + 7a = a – 14 + 7a

2 + 14 = 8a – 14 + 14

16 = 8a

8 = p

2 = a

2 + 5 = 3 - 1

2 + 5 - 2 = 3 – 1 - 2

5 + 1 = – 1 + 1

6 =

3 + 6 – 6 = 12 – 6

3 = 6

6

= 2

39 + 15 = 6c – 15 + 15

54 = 6c

9 = c



c) 2(3y + 1) = 14(y – 1) d) 4(z – 3) = 2(z + 1)

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

__________________________________ __________________________________

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9.8 Solving Problems with Equations

1. A triangle has a perimeter of 50cm. If 2 of its sides are equal and the third side is 5cm

more than the equal sides, what is the length of the third side?

_________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________

2. Three friends share a bag of sweets.

Bill has sweets. Jane has three more sweets than Bill, and Adam has twice as many sweets as Bill.

(a) Write expressions for the number of sweets each child has :

Bill = _________________ Jane = _________________ Adam = _________________

(b) Altogether there are 27 sweets. Write an equation and solve it to find .

________________________________________________________________________________________________________

(c) Write down how many sweets each person has.

________________________________________________________________________________________________________

3. Wayne buys 5 Coke and 7 Kinnie. Bottles of Kinnie cost 60c each.

(a) Write and expression for the cost of the drinks. (use c for the cost of a Coke bottle)

______________________________________________________________________________________

(b) Wayne spends €6.70 on the drinks. How much is a bottle of Coke?

______________________________________________________________________________________

You can

always

check

your

answer

in

algebra!

6y + 2 – 6y = 14y – 14 – 6y

2 + 14 = 8y – 14 + 14

16 = 8y

2 = y

4z – 12 + 12 = 2z + 2 + 12

4z – 2z = 2z + 14 – 2z

2z = 14

z = 7

50 = + + ( + 5) 50 = 3 + 5

50 – 5 = 3 + 5 – 5 45 = 3 = 15

+ 3 2

+ ( + 3) + 2 = 27 4 + 3 – 3 = 27 – 3 4 = 24 = 6

Bill = 6 sweets Jane = 9 sweets Adam = 12 sweets

5c + (7 x 60) = 5c + 420

670 – 420 = 5c + 420 – 420 250 = 5c 50 = c

topic 9: expressions, formulae & equations fileform 2 – expressions, formulae & equations...

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