Collins
•
Master topics and boost confidence with concepts broken down into small steps
•
Learn from visual examples and models that ensure a deep understanding of maths
•
Build your fluency, reasoning and problemsolving skills with purposeful practice
8424_SB2 White Rose Maths Secondary.indd 1
978-0-00-840090-3
Ian Davies and Caroline Hamilton
978-0-00-840088-0
White Rose Maths Key Stage 3 Student Book 2
The mastery programme for KS3 that lays the foundations for GCSE 9–1 success.
White Rose Maths Key Stage 3 Student Book 2
Ian Davies and Caroline Hamilton 01/02/2021 15:01
7.4 Brackets and equations Small steps
Key words
■ Solve equations, including with brackets
Solve – find a value that makes an equation true
■ Form and solve equations with brackets
Solution – a value you can substitute in place of the unknown in an equation to make it true
Are you ready? Solve the equations.
1
x = 40
c
e
x – 40 = 2
f
x + 2 = 40 40 – x = 2
b
4y + 1 = 68
c
4y – 1 = 78
a
2x = 40
b
d
x + 40 = 2
2
Solve the equations.
2
4y = 68
a
78 = 4y – 1
d
Solve the equations.
3
z = 15
a
b
3
z – 4 = 15
3
c
z + 4 = 15
d
3
z – 3 = 15
e
4
z + 3 = 15 4
Expand the brackets and simplify where possible.
4
a
4(x + 3)
b
5(y – 3)
c
7(3 – p)
d
4(a – 3) + 2(3a – 5)
Models and representations In Book 1, you met several different ways of representing equations. Cups and counters
Balance 1
1
1
1
1
1
1
x 1
x 1
x 1
10 1
1
1
1
Algebra tiles
Bar model
x
2
x
2
x
2
18
These all show the equation 3(x + 2) = 18
249
8400_KS3_Maths_SB2.indb 249
23/03/2021 14:55
7.4 Brackets and equations Example 1 Solve the equation 3(x + 2) = 18
Method A You could start by expanding the brackets
–6
3(x + 2) = 18 3x + 6 = 18 3x
Now you have an equation of the form you solved in Book 1.
–6
Subtract 6 from each side to isolate a term in x
÷3
Then divide by 3 to find x
= 12
÷3
x
=4
You can also use bar models or other representations if you prefer.
x
x
2
x
2
2
18
x
x
Collect like terms
x
2 2 2
18
x
x
−6 from both boxes
x
12
Divide into three equal parts
x
x
x
4
4
4
Each x = 4
Method B 3(x + 2) = 18 ÷3
÷3
You could start by dividing by 3
−2
Now subtract 2 from both sides
x+2=6 −2
x
=4
Here is how this method looks using bar models.
x
2
x 18
x
2 6
2
x
2 Divide into three equal parts −2 from both boxes
Remember: you can check the answer to an equation by substituting the answer back into the original equation. 3(x + 2) = 18 Substitute x = 4
x
3(4 + 2 ) = 18
4
3 × 6 = 18, which is correct.
250
8400_KS3_Maths_SB2.indb 250
23/03/2021 14:55
7.4 Brackets and equations Example 2 Solve the equations. a
5(y – 4) = 12
b 30 = 5(7 + t) It is probably best to start with expanding the brackets here, as 5 doesn’t divide exactly into 12
5(y – 4) = 12 5y – 20 = 12
a + 20
5y
To isolate 5y you need to add 20 to both sides.
÷5
Then divide both sides by 5
= 32
÷5
y b
+ 20
The answer is 6.4. Remember, answers don’t have to be integers. You can check by substitution.
= 6.4
30 = 5(7 + t) ÷5
It doesn’t matter that the unknown is on the right-hand side; you can work in the same way. As 30 is divisible by 5 you could use either method easily.
÷5 6=7+t
−7
−7 −1 = t
You can leave the answer as –1 = t or rewrite it as t = –1
Practice 7.4A 1
Compare solving 2x + 5 = 18 with 2(x + 5) = 18
2
Solve the equations. a
4(a + 12) = 60
b
24 = 3(b – 5)
c
6(c + 1.5) = 18
3
What's the same and what's different about solving 3(x + 4) = 15 and 3(2x + 4) = 15?
4
Solve the equations.
5
a
2(3x – 1) + 4x = 12
b
2(3x + 1) – 4x = 12
c
2(3x – 1) – 4 = 12
d
2(3x – 1) + 4 = 12
y + 7 = 19
b
3y + 7 = 19 2
3(y + 7) = 19 2
d
3(y – 7) = 19 2
Solve the equations. a c
6
2
Here are Beca’s and Jackson’s methods for solving the equation 10 – 4x = 8
Beca 10 − 4x = 8 − 10
Jackson 10 − 4x = 8 − 10
+ 4x 10 = 8 + 4x
−4x = −2 ÷ –4
÷ –4
x=1 2
+ 4x
−8
−8 2 = 4x ÷4
÷4 1 2 =x
251
8400_KS3_Maths_SB2.indb 251
23/03/2021 14:55
7.4 Brackets and equations a
Whose method do you prefer?
b
Solve the equations. i 10 – 3x = 7
ii 10 – 3(x – 1) = 7 iv 10 – x + 1 = 7 3
iii 10 – 3(x + 1) = 7
What do you think? 1
Seb thinks of a number. He adds 4 and then multiplies by 5. His result is 70 a
Calling Seb’s number n, which of the equations represents Seb’s number puzzle?
n + 20 = 70 b
2
5n + 4 = 70
5(n + 4) = 70
Solve the equation. Check your answer by substituting back into the steps in the puzzle.
Here are two number puzzles.
I think of a number. I multiply by 2 and add 7. My answer is 23 Faith
Ed I think of a number. I add 7 and multiply by 2. My answer is 23
a
Do you think Ed’s and Faith’s numbers are the same or different? Why?
b
Find Ed’s and Faith’s numbers by i writing them as function machines and working backwards ii forming and solving equations.
c
Find the number Abdullah is thinking of. I think of a number. I multiply by 4, add 8 and divide by 2. My answer is 1
d
Make up your own number puzzles and challenge a partner.
Equations can be formed to help solve puzzles, real-life problems and problems within other areas of mathematics.
252
8400_KS3_Maths_SB2.indb 252
23/03/2021 14:55
7.4 Brackets and equations Example 3 Twice Huda’s age in 3 years’ time is her mother’s age now. Huda’s mother is 36. How old is Huda now?
Let Huda be n years old.
Choose a letter to represent Huda’s age.
In 3 years she will be n + 3 years old.
Work through the information in the question, forming expressions as you go.
Twice this is 2(n + 3) This is equal to her mother’s age. 2(n + 3) = 36
You can then form an equation.
÷2
÷2
Solve the equation. You could expand the brackets first if you prefer.
n + 3 = 18 −3
−3
n = 15
Example 4 The sum of three consecutive numbers is 42. Find the smallest number.
Let the smallest number be x
Choose a letter to represent the smallest number.
The other numbers will be x + 1 and x + 2
Consecutive means following on from each other. One more than x is x + 1, and one more than x + 1 is x + 2
x + (x + 1) + (x + 2) = 42 3x + 3 = 42 −3
−3 3x = 39
÷3
÷3
You can form an equation as you know the sum of the numbers is 42. Notice the brackets are just to make the three numbers clear, and there is nothing to expand.
x = 13 You can check the answer by adding 13, 14 and 15 The sum is 42, so 13 is the smallest number.
Practice 7.4B 1
If you treble the age Rhys was four years ago, you get 27. Form and solve an equation to find Rhys’ age now.
2
If you double the age Ali will be in six years’ time, you will get 40. How old is Ali now?
3
a
In a bag of marbles there are three times as many green marbles as red marbles, and twice as many red marbles as blue marbles. There are 135 marbles in the bag. i In forming an equation, why is it easier to call the number of blue marbles x instead of calling the number of green marbles x? ii Form and solve an equation to find the number of marbles of each colour in the bag.
253
8400_KS3_Maths_SB2.indb 253
23/03/2021 14:55
7.4 Brackets and equations b
In another bag of marbles there are two more white marbles than yellow marbles, and three times as many pink marbles as white marbles. There are 108 marbles in the bag. Calling the number of yellow marbles x, form and solve an equation to find the number of marbles of each colour in the bag.
4
In a shop, there are three more £10 notes than £5 notes. Altogether there is £210 made up of £5 and £10 notes in the till. Compare forming and solving equations to find the number of each type of note if a
5
6
the number of £5 notes is x
b
the number of £10 notes is x
The perimeter of a hexagon is given by the formula P = 3(2x + y) a
Find y if x = 5 and the perimeter is 51 units.
b
Find x if y = 5 and the perimeter is 51 units.
The perimeter of a rectangle is 52 cm. The length is 4 cm greater than the width. Find the area of the rectangle.
What do you think? 1
Amina, Darius and Junaid are playing a game. Junaid scores 4 points more than Darius and Amina scores twice as many points as Junaid. They score 132 points altogether. Find the difference between the number of points scored by Amina and Darius.
2
In how many different ways can you find the values of the letters by forming and solving equations? Which ones involve brackets? What angle facts are you using? a
b 44°
(2x − 15)°
(3x − 5)°
(5x − 2)°
Consolidate – do you need more? 1
2
Solve the equations. a
6x + 3 = 45
b
6(x + 3) = 45
c
6x – 3 = 45
d
6(x – 3) = 45
b
40 = 5(4p – 2)
Solve the equations. a
3
3(2y – 1) = 27
c
6(2q + 7) = 18
Solve the equations. a
3(4a – 2) – 10 = 44
c
4(2 + 3c) – 4(3 + 2c) = 11
b 3(4b – 2) – 10b = 44
254
8400_KS3_Maths_SB2.indb 254
23/03/2021 14:55
7.4 Brackets and equations 4
Flo scores 5 marks more than Emily in a test. Kath scores twice as many marks as Flo. Between them, they score 99 marks. Find their marks in the test.
Stretch – can you deepen your learning? 1
2
Solve the equations. a 2(a – 6) = 2 3 c 1(7 – c) = 3 4 a
b d
2(b + 7) = 4 5 4(5 – 2d) = 1 3
What's the same and what's different about solving these equations? 2n + 3 = 4
2n + 3 = 4n
You will explore solving equations like 2n + 3 = 4n, where the letter appears on both sides of the equation, in Chapter 7.5 b 3
Compare methods for solving the equations 2(n + 3) = 4 and 2(n + 3) = 4n
Form and solve equations to solve these number puzzles. a
I think of a number, double it and add 5. The answer is the number I started with.
b
I think of a number, add 5 and then double the result. The answer is the number I started with.
Reflect 1
How do you go about solving an equation with brackets?
2
How do you use the information in a question to form an equation?
3
How can you check if your answer to an equation is correct?
255
8400_KS3_Maths_SB2.indb 255
23/03/2021 14:55
Collins
•
Master topics and boost confidence with concepts broken down into small steps
•
Learn from visual examples and models that ensure a deep understanding of maths
•
Build your fluency, reasoning and problemsolving skills with purposeful practice
8424_SB2 White Rose Maths Secondary.indd 1
978-0-00-840090-3
Ian Davies and Caroline Hamilton
978-0-00-840088-0
White Rose Maths Key Stage 3 Student Book 2
The mastery programme for KS3 that lays the foundations for GCSE 9–1 success.
White Rose Maths Key Stage 3 Student Book 2
Ian Davies and Caroline Hamilton 01/02/2021 15:01