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MODULE 7 WORKBOOKForming and Solving Equations
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7 May 2019
Contents
Lesson 1 p 3
Lesson 2 p 9
Lesson 3 p 16
Apply (Non-calculator) p 23
Apply (Calculator) p 28
2
Module 7
Forming and solving equations PODS Before continuing, make sure you have watched and have access to the following Pods.To find these, simply log in to GCSEPod and enter the codes given into the search bar (orclick on the titles below).
1. Linear equations in one unknown | MATHS-04-001
2. Simultaneous equations (linear only) | MATHS-04-005
3. Linear inequities | MATHS-04-010
Lesson 1
Solving linear equations
PODS
1. Linear equations in one unknown | MATHS-04-001
Quiz
Watch the pod and answer the following questions.
1. What is the correct solution to the equation 2x + 7 = 19?
a) x = 6
b) x = 13
c) x = 12
d) x = 2.5
………………………………………………………………………………………………………………………………………………………………..
3
2. Which of the following is not a true statement?
a) 4 x -3 = -8 – 4
b) !"
x -8 = 7 + -9
c) 3 + -7 = 2 x -4
d) 8 ÷ -2 = -10 + 6
………………………………………………………………………………………………………………………………………………………………..
3. Which value of x is different from the others?
a) 5x + 4 = -6
b) $%+ 3 = 2
c) 2(x + 7) = 10
d) -2 = -x
………………………………………………………………………………………………………………………………………………………………..
4. Solve the equation 3 – 2x = 9.
a) x = 3
b) x = -3
c) x = 6
d) x = -6
………………………………………………………………………………………………………………………………………………………………..
5. Find the value of x when 7 = $%
a) x = 3.5
b) x = %+
c) x = -14
d) x = 14
………………………………………………………………………………………………………………………………………………………………..
4
6. Solve the following equation to find the value of x: 9x – 2 = 2x + 12
a) x = 2
b) x = 1
c) x = -2
d) x = !,+
………………………………………………………………………………………………………………………………………………………………..
7. Find the value of the unknown in the equation 4p + 2 = 8
a) p = 0
b) p = 2.3
c) p = 1.5
d) p = 4
………………………………………………………………………………………………………………………………………………………………..
8. Solve the equation 3(2x – 3) = 6
a) x = 2.5
b) x = 10.5
c) x = 1
d) x = 2
………………………………………………………………………………………………………………………………………………………………..
9. Solve 2"+ 3 = 7
a) y = 40
b) y = 1
c) y = 25
d) y = 16
………………………………………………………………………………………………………………………………………………………………..
10. Calculate the correct solution to the equation -5x = -45
a) x = -9
b) x = 9
c) x = !3
d) x = - !3
………………………………………………………………………………………………………………………………………………………………..
5
Practise
Practise solving linear equations.
Do not use a calculator.
1. Solve -4(p + 7) = 4
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
2. Solve $%
– 3 = 3x
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
3. Solve "5
= 3
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
4. Solve !%(5𝑥 + 12) = !
"(6𝑥 + 8)
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
5. Solve 4 >$%+ 1? = 3>3 − $
A?
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
6
6. The diagram below shows a square.
Given that the perimeter of the square is 1 metre, calculate the value of x.
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
7. Claire says “I think of a number, multiply it by 3 and then add on 4. The answer I get is13.”
Write an equation to represent this statement.
………………………………………………………………………………………………………………………………………………………………..
8. Paul puts some biscuits in a tin. He then eats half of them. He cannot resist and eatsanother 3. He then counts how many biscuits he has left which is 7. How many biscuitsdid he put in the tin?
………………………………………………………………………………………………………………………………………………………………..
9. Molly is x years old. Suzy is 4 years older. The sum of their ages is 24. How old is
Molly?
………………………………………………………………………………………………………………………………………………………………..
7
10. The diagram shows a straight line.
Calculate the value of x.
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
8
Lesson 2
Simultaneous equations
PODS
2. Simultaneous equations (linear only) | MATHS-04-005
Quiz
Watch the pod and answer the following questions.
1. Which of the following operations is needed to make this sentence true:
a) +
b) -
c) x
d) ÷
………………………………………………………………………………………………………………………………………………………………..
2. Which of the following equations is not equivalent to the others:
a) 2x – 3y = 7
b) 2x = 7 + 3y
c) 0 = 3y – 2x + 7
d) 3x + 3y – 7 = 0
………………………………………………………………………………………………………………………………………………………………..
9
3. Which of the following equations is not equivalent to 4x + 3y = -1?
a) 12x + 9y = -3
b) 16x + 12y = -4
c) -8x – 6y = 2
d) 8x + 6y = 2
………………………………………………………………………………………………………………………………………………………………..
4. I want to eliminate y from the equations 3x + y = 4 and 7x – y = 1. The quickest process
which will do this is:a) Adding the two equations together.b) Taking one away from the other.
c) Rearrange one equation to get y = and then substitute it into the other.
d) Manipulate so that both equations have the same number of x terms and then
subtract.
………………………………………………………………………………………………………………………………………………………………..
5. The lowest common multiple of 3x and 2x is
a) 6x2
b) 5x
c) 6xd) 1.5
………………………………………………………………………………………………………………………………………………………………..
6. If 2y = 3x +4, then which of the following statements is not true:
a) y = 1.5x + 2
b) 4y = 6x + 8c) %
A𝑦 = 𝑥 + "
A
d) 2y + 3x = 4
………………………………………………………………………………………………………………………………………………………………..
7. If the equation 2y – 3x = 7 is multiplied by -2, then the result is:
a) -4y – 6x = -14
b) -4y + 6x = -14
c) -4y + 6x = 14
d) 4y -6x = 14
………………………………………………………………………………………………………………………………………………………………..
10
8. Calculate the value of the star and the circle when:
a) Star = 12, Circle = 11
b) Star = 6, Circle = 1
c) Star = 6, Circle = 5
d) Star = 6, Circle = 1
………………………………………………………………………………………………………………………………………………………………..
9. Calculate the value of the star and the circle when:
a) Star = -9, Circle = 8
b) Star = 9, Circle = 8
c) Star = -9, Circle = 10
d) Star = -9, Circle = -10
………………………………………………………………………………………………………………………………………………………………..
10. 3a = 15 and a + b = 2. Calculate the values of a and b.
a) a = 5, b = 3
b) a = 3, b = -1
c) a = 5, b = 7
d) a = 5, b = -3
………………………………………………………………………………………………………………………………………………………………..
11
Practise
Practise simultaneous equations.
Do not use a calculator.
1. Solve the following simultaneous equations:
2x + y = 5
x – y = 7
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
2. Solve the following simultaneous equations:
3x – 2y = 6
x + y = 7
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
3. Solve the following simultaneous equations:
5x + 2y = -6
x + 2y = 2
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
4. Solve the following simultaneous equations:
3x + 2y = 9
2x – 3y = -7
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
12
5. Solve the following simultaneous equations:
y = 2x + 3
y = 5 – 6x
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
13
Practise
Practise forming and solving equations.
You may use a calculator.
1. Solve the following simultaneous equations:
y = 2x + 5
y = 16 – 4x
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
2. Solve the following simultaneous equations:
y = 2x – 1
2y + 3x = 7
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
3. 3 apples and 4 bananas cost £1.88. 5 apples and 3 bananas cost £2.07.
Calculate the cost of 1 apple and 1 banana.
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
4. Sarah and Claire are in a café. Sarah buys 2 coffees and 3 biscuits for £5.85. Claire thenbuys 3 coffees and 1 biscuit for £6.15.
Calculate the cost of 1 coffee and 1 biscuit.
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
14
5. Solve the following simultaneous equations:
2x – 3y = -1.8
5x – 2y = -2.3
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
15
Lesson 3
Inequalities
PODS
3. Linear inequities | MATHS-04-010
Quiz
Watch the pod and answer the following questions.
1. Which inequality is shown on the number line below?
a) x ≥-3
b) x >3
c) x <3
d) x ≤-3
………………………………………………………………………………………………………………………………………………………………..
2. Which inequality is shown on the number line below?
a) x ≤-1
b) x <-1
c) x >-5
d) -5<x <-1
………………………………………………………………………………………………………………………………………………………………..
16
3. Which inequality is shown on the number line below?
a) -4≤x <2
b) -4<x <2
c) -4≤x ≤2
d) -4≤x <4
………………………………………………………………………………………………………………………………………………………………..
4. Which number line represents the inequality -2≤x <3?
………………………………………………………………………………………………………………………………………………………………..
5. The solution to the inequality 3x + 4 ≥ 10 is:
a) x ≥6
b) x ≥2
c) x ≥-2
d) x ≤2
………………………………………………………………………………………………………………………………………………………………..
a) c)
b) d)
17
6. Which is the correct solution to the inequality 4x–3≤ 3x–2?
a) c)
b) d)
………………………………………………………………………………………………………………………………………………………………..
7. Solve -3x >12.
a) x <-4
b) x >-4
c) x <4
d) x >4
………………………………………………………………………………………………………………………………………………………………..
8. Solve 3 – 2x >8.
a) x >-2.5
b) x >2.5
c) x <2.5
d) x <-2.5
………………………………………………………………………………………………………………………………………………………………..
9. Which of the following is not a correct manipulation of the inequality -2 < 2x – 1 <10?
a) -1 <2x <11
b) -0.5 <x <5.5
c) -1 <x – 1 <5
d) -1 <x – 0.5 <5
………………………………………………………………………………………………………………………………………………………………..
18
10. Which list of numbers is valid for both of the inequalities shown?
a) -3, -2, -1, 0, 1, 2, 3
b) -1, 0, 1
c) -1, 0, 1, 2
d) -2, -1, 0, -1
………………………………………………………………………………………………………………………………………………………………..
19
Practise
Practise inequalities.
Do not use a calculator.
1. Solve the inequality 2x – 3 > 3x + 7
………………………………………………………………………………………………………………………………………………………………..
2. Write down all of the values of x which satisfy the following inequalities:
a) x – 3 < 5 where x is a positive integer
………………………………………………………………………………………………………………………………………………………………..
b) 5x + 1 < 2x where x is an integer and x ≥-5
………………………………………………………………………………………………………………………………………………………………..
c) 3(2x – 8) < 10 where x is a positive square number
………………………………………………………………………………………………………………………………………………………………..
3. Solve the inequality 4(3 – 2x) < 2(3x -7)
………………………………………………………………………………………………………………………………………………………………..
4. The perimeter of a square is greater than 10cm and less than 20cm. Given that the sidelength of the square is an integer, find the minimum and maximum areas of thesquare.
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
20
5. Write down the inequality which satisfies both of the following:
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
21
Practise
Practise inequalities.
You may use a calculator.
1. Solve the inequality 𝟏𝟐
3x – 4 ≥2(2 – 2x).
………………………………………………………………………………………………………………………………………………………………..
2. Solve the 2 inequalities and draw the solution set on the number line below.
4x – 7 > 3 and 6 – 3x > -12
………………………………………………………………………………………………………………………………………………………………..
3. Solve the inequality 17x – 52 > 23 – 4x giving your answer correct to 2 decimal places.
………………………………………………………………………………………………………………………………………………………………..
4. Solve the inequality - 𝟑𝟒
x >7.
………………………………………………………………………………………………………………………………………………………………..
5. By solving both sides of this inequality, find a range of solutions for x: -3 <2x + 5≤ 6
………………………………………………………………………………………………………………………………………………………………..
22
Apply
Apply what you have revised about forming and solving equations.
Do not use a calculator.
Remember
Solving equations is like using a balance; always do the same thing to both sides of the equals sign.
Be extremely careful with negatives – remember to perform the opposite operation to “undo” an equation.
When solving simultaneous equations, you are aiming to find equivalent equations with the same quantity of x OR y terms so that you can ELIMINATE them. Find the second solution at the end of the question.
When working with inequalities you must reverse the sign if you multiply or divide by a negative.
1. Calculate the value of x in this rectangle.
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
23
2. The diagram shows an equilateral triangle and a regular hexagon. Both shapes have
the same perimeter. Calculate the value of x.
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
3. In the rectangle below, calculate the length of the shortest side and comment on yourresult.
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
24
4. Calculate the value of x in the triangle below.
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
5. The length of a paddling pool is 3 times the width. If the perimeter of the pool is 16m,calculate its area.
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
6. Calculate the area of this rectangle.
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
25
7. A class go on a trip to a cinema. They buy 10 children’s tickets and 2 adult tickets for£28. More children decide to come on the trip, and so they buy an additional 8children’s tickets and 1 more adult ticket for £20.
Calculate the cost of a children’s ticket and an adult ticket.
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
8. The diagram shows an equilateral triangle and a regular hexagon.
Find the range of values for which the perimeter of the triangle is greater than the perimeter of the hexagon.
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
9. Bob is x years old. Freddie is 3 years older than Bob. Clara is twice as old as Freddie.
Dave is 6 times as old as Bob. The sum of Bob, Freddie and Clara’s ages is less than theage of Dave.Find the range of ages that Bob can be.
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
26
10. The diagram below shows a function machine.
a) Calculate the output when the input is 6.
………………………………………………………………………………………………………………………………………………………………..
b) Calculate the output when the input is -2.
………………………………………………………………………………………………………………………………………………………………..
c) Calculate the input when the output is 7.
………………………………………………………………………………………………………………………………………………………………..
d) Write out the inverse function machine.
………………………………………………………………………………………………………………………………………………………………..
27
Apply
Apply what you have revised about forming and solving equations.
You may use a calculator.
Remember
Solving equations is like using a balance; always do the same thing to both sides of the equals sign.
Be extremely careful with negatives – remember to perform the opposite operation to “undo” an equation.
When solving simultaneous equations, you are aiming to find equivalent equations with the same quantity of x OR y terms so that you can ELIMINATE them. Find the second solution at the end of the question.
When working with inequalities you must reverse the sign if you multiply or divide by a negative.
1. The length of a paddling pool is 4 times the width. If the perimeter of the pool is 27m,calculate its area.
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
28
2. The diagram below shows a square.
Given that the perimeter of the square is 1.4 metres, calculate the value of x
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
3. I think of a number, multiply it by 3 and then add on 14. The answer I get is -2. What isthe number I was thinking of?
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
4. Calculate the size of the largest angle in this triangle.
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
29
5. The diagram below shows a straight line. Calculate the size of each angle.
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
6. A class go on a trip to a cinema. They buy 30 children’s tickets and 4 adult tickets for£108. More children decide to come on the trip, and so they buy an additional 40children’s tickets and 6 more adult tickets for £147.
Calculate the cost of a children’s ticket and an adult’s ticket.
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
7. Solve the following simultaneous equations:
14x + 5y = 5
6x – 2y = 0.9
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
30
8. Simon works in a factory. He gets paid a standard hourly wage plus extra for everyhour of overtime he works.
One week, Simon works for 10 hours at the standard rate and does 3 hours ofovertime; he receives £178.
The next week Simon works for 15 hours at the standard rate and does 5 hours ofovertime; he receives £276.25.
Calculate the amount of money that Simon receives for each hour of overtime that heworks.
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
9. Jenny solves the inequality in class as follows:
Is Jenny correct? Give reasons for your answer.
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
31
10. The rectangle below has a perimeter greater than 20cm and less than 32cm. Given
that x is an integer, calculate the area of the rectangle.
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
………………………………………………………………………………………………………………………………………………………………..
32
