3(a + 5)What does this mean?

‘add five to a then multiply the whole lot by three’

Or ‘three lots of a added to three lots of 5

Expanding Brackets

3(a + 5)

Expanding Brackets

+ 5

+ 5

+ 5a

a

a

3(a + 5)

Expanding Brackets

+ 5

+ 5

+ 5a

a

a

3(a + 5) =

3(a + 5)

Expanding Brackets

+ 5

+ 5

+ 5a

a

a

3(a + 5) = (3 x a) +

3(a + 5)

Expanding Brackets

+ 5

+ 5

+ 5a

a

a

3(a + 5) = (3 x a) + (3 x 5) =

3(a + 5)

Expanding Brackets

+ 5

+ 5

+ 5a

a

a

3(a + 5) = (3 x a) + (3 x 5) = 3a + 15

Expanding Brackets6(2a + 4)

+ 4

+ 4

+ 4

6(2a + 4) =

+ 4 + 4

+ 4

(6 x 2a) + (6 x 4)= 12a + 24

Example:5(2z – 3)

Each term inside the brackets is multiplied by the number outside the brackets.

Watch out for the signs!

Expanding Brackets

Example:5(2z – 3)

(5 x 2z) + 5 x -3

Expanding Brackets

Example:5(2z – 3)

(5 x 2z) + 5 x - 3

= 10z – 15

Expanding Brackets

Example:2(3p + 4) + 3(4p + 1)

Expanding Brackets

Example:2(3p + 4) + 3(4p + 1)

= (2 x 3p) + (2 x 4)

Expanding Brackets

Example:2(3p + 4) + 3(4p + 1)

= (2 x 3p) + (2 x 4) + (3 x 4p) + (3 x 1)

Expanding Brackets

Example:2(3p + 4) + 3(4p + 1)

= (2 x 3p) + (2 x 4) + (3 x 4p) + (3 x 1)

= 6p + 8

Expanding Brackets

Practice 1: Expand the brackets:

(a) (i) 7(n – 3)

(ii) 4(2x – 3)

  (iii) p(q – 2p) 

Multiply out: (3)

(a) 5(2y – 3)

(1)(c) x(2x +y)

(2)Lesson

7n - 21

8x -12

pq – 2p²

10y - 15

2x² + xy

Are you ready for the answers ?

Practice 2: Expand and simplify:

(i) 4(x + 5) + 3(x – 7)

(2) (ii) 5(3p + 2) – 2(5p – 3)

(2)

(2)

Lesson

4x + 9 + 3x -21 = 7x - 12

15p + 10 - 10p + 6 = 5p +16

Are you ready for the answers ?

By using substitution answer the following questions:

(i) Work out the value of 2a + ay when a = 5 and y = –3  

(2)(ii) Work out the value of 5t² - 7 when t=4

(iii) Work out the value of 5x + 1 when x = –3

(iv) Work out the value of D when: (4)D = ut + 2kt

If u = 5t = 1.2k = –2

(3)Lesson

-5

73

-14

1.2

Are you ready for the answers ?

TOP How much do you know?Solve the following

(i) x + 5 = 16

(ii) 3x + 4 = 19

(2)(b) 6y + 9 = 45

(1)

(c) 2x – 5 = -1 (2)

(d) 4(x + 3) = 20 (1)

(e) 29 = 9x - 7 (1)

(Total 7 marks)

Lesson

x = 11

x = 5

y = 6

x = 2

x = 4

x = 2

Are you ready for the answers ?

Practice 1: Solve:

(a) (i) 4x + 2 = 18

(ii) 8x – 5 = 19

  (iii) 7 = 3y - 8 

Multiply out the brackets first:

(a) 2(x + 3) = 16

(1)(c) 3(2x – 3) = 9

(2)Lesson

x = 3

y = 5

x = 5

x = 3

x = 4

Are you ready for the answers ?

Practice 2: Solve:

(i) 2x + 3 = x + 7

(2) (ii) 8r + 3 = 5r + 12

(2) (iii) 9x – 14 = 4x + 11

(2)

(iv) 20y – 16 = 18y - 10

(2)

Lesson

x = 4

3r = 9 r = 3

5x = 25 x = 5

2y = 6 y = 3

Are you ready for the answers ?

Crossing the equals sign

When we take a value across the equals sign we change what it was doing to the opposite.

So, if it was + 2 on one side, when we take it to the other it is – 2If we are x 2 on one side, when we take it to the other it is / 2

For example,

x + 5 = 13

x = 13 – 5

x = 8

Using inverse operations to solve equations

Solve the following equations using inverse operations.

5x = 45

x = 45 ÷ 5x = 9

Check:

5 × 9 = 45

17 – x = 6

17 = 6 + x17 – 6 = x

Check:

17 – 11 = 6

11 = xx = 11

We usually write the letter before the equals sign.

Using inverse operations to solve equations

Solve the following equations using inverse operations.

x = 3 × 7x = 21

Check:

3x – 4 = 14

3x = 14 + 43x = 18

Check:

3 × 6 – 4 = 14

x = 18 ÷ 3x = 6

= 3x

7

= 3217

Balancing equations

Constructing an equation

Ben and Lucy have the same number of sweets.

Ben started with 3 packets of sweets and ate 11 sweets. Lucy started with 2 packets of sweets and ate 3 sweets.

How many sweets are there in a packet?

Let’s call the number of sweets in a packet, n.

We can solve this problem by writing the equation:

3n – 11

The number of Ben’s sweets

=

is the same as

the number of Lucy’s sweets.

2n – 3

Solving the equation

Move the unknowns (letter terms) to one side and the numbers to the other

3n – 11 = 2n – 3 Start by writing the equation down.

3n – 2n = –3 + 11

This is the solution.

We can check the solution by substituting it back into the original equation:

3 8 – 11 = 2 8 – 3

3n - 2n – 11 = – 3

n = 8