Solving Equations: Fluency and Exam Drill (Part 2) - Worksheets, Questions and Revision

26 original exam-style questions - 6 pages of questions with a full mark scheme - free printable PDF.

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GCSE · Algebra

3.14DB Solving Equations: Fluency and Exam Drill (Part 2)

EDEXCEL 1MA1 · Calculators not allowed · about 90 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Solve 8 + x = 15

(Total for Question 1 is 1 mark)

Solve 20 - x = 6

(Total for Question 2 is 1 mark)

Solve 36 = 4x

(Total for Question 3 is 1 mark)

Solve 8 = x / 7

(Total for Question 4 is 1 mark)

A function machine multiplies a number by 3, then adds 4. The output is 19.
Work out the input number.

(Total for Question 5 is 2 marks)

Solve 4x + 3 = 23
Show your working.

(Total for Question 6 is 2 marks)

Solve 6x - 5 = 19
Show your working.

(Total for Question 7 is 2 marks)

Solve 3x + 8 = x + 20
Show your working.

(Total for Question 8 is 2 marks)

Solve 7x - 4 = 3x + 24
Show your working.

(Total for Question 9 is 2 marks)

Solve 5(x - 2) = 25
Show your working.

(Total for Question 10 is 2 marks)

Solve 4(3x + 1) = 28
Show your working.

(Total for Question 11 is 2 marks)

Solve (x - 9) / 5 = 4
Show your working.

(Total for Question 12 is 2 marks)

Solve 9 - 4x = -7
Show your working.

(Total for Question 13 is 2 marks)

Solve 12 - 3x = 4x - 9
Show your working.

(Total for Question 14 is 2 marks)

Solve x / 4 - 3 = 2
Show your working.

(Total for Question 15 is 2 marks)

Three consecutive integers have a sum of 72. Using n for the smallest integer, form an equation and solve it to find the three integers.

(Total for Question 16 is 3 marks)

Yusuf is solving the equation 3(x - 4) = 21. His working is shown below.
3x - 4 = 21
3x = 25
x = 25 / 3

(a)Identify the error in Yusuf's working.(1)

(b)Write out a correct solution to the equation.(2)

(Total for Question 17 is 3 marks)

Solve 5(x + 3) = 2(x + 12)
Show your working.

(Total for Question 18 is 3 marks)

Solve 6(2x - 1) - 4(x - 3) = 22
Show your working.

(Total for Question 19 is 3 marks)

The diagram shows a triangle with its three interior angles marked in terms of x.

Figure (to be drawn): A triangle with each of its three interior angles labelled: one angle is (3x + 5) degrees, another is (2x - 5) degrees, and the third is (x + 30) degrees.

(a)Form an equation in x using the fact that the angles in a triangle sum to 180 degrees, and solve it to find the value of x.(3)

(b)Work out the size of each of the three angles, and state whether the triangle is acute-angled, right-angled or obtuse-angled.(3)

(Total for Question 20 is 6 marks)

The diagram shows a rectangular photo frame. The perimeter of the frame is 50 cm.

Figure (to be drawn): A rectangle representing a photo frame, with the width labelled (x + 2) cm on one of the shorter sides and the length labelled (3x - 1) cm on one of the longer sides.

(a)Show that 8x + 2 = 50(2)

(b)Solve the equation to find the value of x.(2)

(c)Work out the length and the width of the frame, and state whether the frame is in landscape or portrait orientation.(2)

(Total for Question 21 is 6 marks)

Amara thinks of a number and calls it x. She multiplies it by 4, then subtracts 7. Her result is the same as when she adds 5 to the number and then doubles it.
Form an equation in x and solve it to find the number Amara is thinking of.

(Total for Question 22 is 3 marks)

Solve (3x + 2) / 4 - (x - 1) / 3 = 2
Show your working.

(Total for Question 23 is 3 marks)

Solve (5x - 3) / 4 = (3x + 1) / 2
Show your working.

(Total for Question 24 is 3 marks)

Solve 3 / (x + 2) = 2 / (x - 1)
Show your working.

(Total for Question 25 is 3 marks)

Show that the equation (2x + 1) / 3 = (2x - 5) / 3 + 4 has no solutions.

(Total for Question 26 is 3 marks)

Mark scheme · 3.14DB Solving Equations: Fluency and Exam Drill (Part 2)

Question 1

B1 x = 7 cao
Answer: x = 7

Question 2

B1 x = 14 cao
Answer: x = 14

Question 3

B1 x = 9 cao
Answer: x = 9

Question 4

B1 x = 56 cao
Answer: x = 56

Question 5

M1 3n + 4 = 19 oe, or reverses the machine correctly, e.g. (19 - 4) / 3
A1 input = 5 cao
Answer: 5

Question 6

M1 4x = 20 oe
A1 x = 5 cao
Answer: x = 5

Question 7

M1 6x = 24 oe
A1 x = 4 cao
Answer: x = 4

Question 8

M1 2x + 8 = 20 or 2x = 12 oe (collect x terms correctly)
A1 x = 6 cao
Answer: x = 6

Question 9

M1 4x - 4 = 24 or 4x = 28 oe (collect x terms correctly)
A1 x = 7 cao
Answer: x = 7

Question 10

M1 5x - 10 = 25 oe (expand correctly)
A1 x = 7 cao
Answer: x = 7

Question 11

M1 12x + 4 = 28 oe (expand correctly)
A1 x = 2 cao
Answer: x = 2

Question 12

M1 x - 9 = 20 oe
A1 x = 29 cao
Answer: x = 29

Question 13

M1 -4x = -16 or 4x = 16 oe
A1 x = 4 cao
Answer: x = 4

Question 14

M1 21 = 7x or 12 + 9 = 4x + 3x oe (collect x terms onto one side)
A1 x = 3 cao
Answer: x = 3

Question 15

M1 x / 4 = 5 oe
A1 x = 20 cao
Answer: x = 20

Question 16

M1 n + (n + 1) + (n + 2) = 72 oe, leading to 3n + 3 = 72
A1 n = 23 cao
B1 states all three integers 23, 24 and 25 (ft their n)
Answer: 23, 24 and 25

Question 17

(a) B1 identifies that Yusuf did not multiply the 4 inside the bracket by 3 when expanding, oe (he should have 3x - 12 = 21, not 3x - 4 = 21)
(a) Answer: Yusuf expanded the bracket incorrectly: he multiplied 3 by x but not by 4.
(b) M1 3x - 12 = 21 oe (correct expansion)
(b) A1 x = 11 cao
(b) Answer: x = 11

Question 18

M1 5x + 15 = 2x + 24 (expand both brackets correctly)
dM1 3x = 9 oe (dep, collect terms correctly)
A1 x = 3 cao
Answer: x = 3

Question 19

M1 12x - 6 - 4x + 12 = 22 (expand both brackets correctly, with correct signs)
dM1 8x + 6 = 22 (dep, simplify the left-hand side)
A1 x = 2 cao
Answer: x = 2

Question 20

(a) M1 (3x + 5) + (2x - 5) + (x + 30) = 180 oe (angles in a triangle sum to 180 degrees)
(a) dM1 6x + 30 = 180 leading to 6x = 150 (dep)
(a) A1 x = 25 cao
(a) Answer: x = 25
(b) M1 substitutes x = 25 into each expression (ft their x)
(b) A1 80 degrees, 45 degrees and 55 degrees (all three correct)
(b) B1 correctly states the triangle is acute-angled, since all three angles are less than 90 degrees
(b) Answer: 80 degrees, 45 degrees and 55 degrees; the triangle is acute-angled

Question 21

(a) M1 2[(x + 2) + (3x - 1)] = 50 oe (correct perimeter expression set equal to 50)
(a) A1 correctly expands and simplifies to 8x + 2 = 50 (dep)
(a) Answer: 8x + 2 = 50 (shown)
(b) M1 8x = 48 oe
(b) A1 x = 6 cao
(b) Answer: x = 6
(c) M1 substitutes x = 6 into both expressions (ft their x)
(c) A1 length = 17 cm and width = 8 cm (both required, cao), and correctly identifies landscape orientation since the length is greater than the width
(c) Answer: length = 17 cm, width = 8 cm; landscape orientation

Question 22

M1 4x - 7 = 2(x + 5) oe (correct equation formed from the description)
dM1 expands and collects terms to reach 2x = 17 oe (dep)
A1 x = 8.5 oe cao (accept 17/2), with the conclusion that the number is 8.5
Answer: x = 8.5; the number is 8.5

Question 23

M1 multiplies every term by 12, the LCM of 4 and 3, e.g. 3(3x + 2) - 4(x - 1) = 24 oe
dM1 expands and collects terms to reach 5x + 10 = 24 oe (dep)
A1 x = 2.8 oe cao (accept 14/5)
Answer: x = 2.8

Question 24

M1 cross multiplies correctly, 2(5x - 3) = 4(3x + 1) oe
dM1 expands both sides and collects terms, 10x - 6 = 12x + 4 leading to -2x = 10 (dep)
A1 x = -5 cao
Answer: x = -5

Question 25

M1 cross multiplies correctly, 3(x - 1) = 2(x + 2) oe
dM1 expands both sides and collects terms, 3x - 3 = 2x + 4 leading to x = 7 (dep)
A1 x = 7 cao
Answer: x = 7

Question 26

M1 combines the right-hand side into a single fraction, (2x - 5) / 3 + 4 = (2x + 7) / 3 oe
M1 equates numerators over the common denominator, 2x + 1 = 2x + 7 oe
B1 correct conclusion with valid reason (dep on both M marks), e.g. '2x + 1 = 2x + 7 gives 1 = 7, which is false for every value of x, so the equation has no solutions' oe
Answer: The equation has no solutions, since simplifying gives 1 = 7, which is never true.

Question 25

3 marks

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Question 26

3 marks

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Solving Equations: Fluency and Exam Drill (Part 2) - Worksheets, Questions and Revision

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3.14DB Solving Equations: Fluency and Exam Drill (Part 2)

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