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GCSE · Algebra

3.14D Solving Equations: Fluency and Exam Drill

EDEXCEL 1MA1 · Calculators not allowed · about 90 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Solve x + 7 = 12

(Total for Question 1 is 1 mark)

Solve x - 5 = 9

(Total for Question 2 is 1 mark)

Solve 3x = 18

(Total for Question 3 is 1 mark)

Solve x / 4 = 6

(Total for Question 4 is 1 mark)

Solve 2x + 5 = 17
Show your working.

(Total for Question 5 is 2 marks)

Solve 3x - 7 = 14
Show your working.

(Total for Question 6 is 2 marks)

Solve 5x + 2 = 2x + 14
Show your working.

(Total for Question 7 is 2 marks)

Solve 4x - 3 = 2x + 9
Show your working.

(Total for Question 8 is 2 marks)

Solve 2(x + 3) = 16
Show your working.

(Total for Question 9 is 2 marks)

Solve 3(2x - 1) = 15
Show your working.

(Total for Question 10 is 2 marks)

Solve (x + 6) / 4 = 3
Show your working.

(Total for Question 11 is 2 marks)

Solve 5 - 2x = 1
Show your working.

(Total for Question 12 is 2 marks)

Solve 7 - x = 2x + 1
Show your working.

(Total for Question 13 is 2 marks)

Solve x / 3 + 4 = 9
Show your working.

(Total for Question 14 is 2 marks)

Solve 2(x - 5) = 3(x - 8)
Show your working.

(Total for Question 15 is 3 marks)

Solve 4(x + 1) - 2(x - 3) = 20
Show your working.

(Total for Question 16 is 3 marks)

A rectangle has length (2x + 5) cm and width (x + 3) cm. The perimeter of the rectangle is 46 cm.

(a)Show that 6x + 16 = 46(2)

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

(c)Work out the length and the width of the rectangle.(2)

(Total for Question 17 is 6 marks)

The angles (2x + 20) degrees, (3x + 10) degrees and (x + 30) degrees lie on a straight line.

(a)Form an equation in x and solve it to find the value of x.(3)

(b)Work out the size of each of the three angles, and state which one is the largest.(3)

(Total for Question 18 is 6 marks)

Ben is x years old. His sister Priya is 4 years older than Ben. Their mother's age is 8 years less than three times the sum of Ben's and Priya's ages. Their mother is 52 years old.
Form an equation in x and solve it to find Ben's age.

(Total for Question 19 is 4 marks)

A mobile phone company offers two data plans. Plan A costs a fixed 15 pounds per month plus 2 pounds per gigabyte (GB) of data used. Plan B costs a fixed 9 pounds per month plus 3.50 pounds per GB of data used. Let g be the number of GB used in a month.

(a)Write an expression for the total monthly cost of Plan A, and an expression for the total monthly cost of Plan B.(2)

(b)Form an equation and solve it to find the value of g for which the two plans cost the same amount.(3)

(c)Work out the monthly cost when the two plans cost the same amount.(1)

(Total for Question 20 is 6 marks)

Solve (2x - 1) / 3 + (x + 1) / 2 = 6
Show your working.

(Total for Question 21 is 3 marks)

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

(Total for Question 22 is 3 marks)

Show that the equation 6(x + 2) - 2x = 4(x + 5) has no solutions.

(Total for Question 23 is 3 marks)

Find the value of k for which the equation 3(x + 4) = kx + 12 has an infinite number of solutions.

(Total for Question 24 is 3 marks)

Mark scheme · 3.14D Solving Equations: Fluency and Exam Drill

Question 1

B1 x = 5 cao
Answer: x = 5

Question 2

B1 x = 14 cao
Answer: x = 14

Question 3

B1 x = 6 cao
Answer: x = 6

Question 4

B1 x = 24 cao
Answer: x = 24

Question 5

M1 2x = 12 oe
A1 x = 6 cao
Answer: x = 6

Question 6

M1 3x = 21 oe
A1 x = 7 cao
Answer: x = 7

Question 7

M1 3x + 2 = 14 or 3x = 12 oe (collect x terms correctly)
A1 x = 4 cao
Answer: x = 4

Question 8

M1 2x - 3 = 9 or 2x = 12 oe (collect x terms correctly)
A1 x = 6 cao
Answer: x = 6

Question 9

M1 2x + 6 = 16 oe (expand correctly)
A1 x = 5 cao
Answer: x = 5

Question 10

M1 6x - 3 = 15 oe (expand correctly)
A1 x = 3 cao
Answer: x = 3

Question 11

M1 x + 6 = 12 oe
A1 x = 6 cao
Answer: x = 6

Question 12

M1 -2x = -4 or 2x = 4 oe
A1 x = 2 cao
Answer: x = 2

Question 13

M1 7 = 3x + 1 or 6 = 3x oe (collect x terms onto one side)
A1 x = 2 cao
Answer: x = 2

Question 14

M1 x / 3 = 5 oe
A1 x = 15 cao
Answer: x = 15

Question 15

M1 2x - 10 = 3x - 24 (expand both brackets correctly)
dM1 14 = x oe, i.e. -x = -14 (dep, collect terms correctly)
A1 x = 14 cao
Answer: x = 14

Question 16

M1 4x + 4 - 2x + 6 = 20 (expand both brackets correctly, with correct signs)
dM1 2x + 10 = 20 (dep, simplify the left-hand side)
A1 x = 5 cao
Answer: x = 5

Question 17

(a) M1 2[(2x + 5) + (x + 3)] = 46 oe (correct perimeter expression set equal to 46)
(a) A1 correctly expands and simplifies to 6x + 16 = 46 (dep)
(a) Answer: 6x + 16 = 46 (shown)
(b) M1 6x = 30 oe
(b) A1 x = 5 cao
(b) Answer: x = 5
(c) M1 substitutes x = 5 into both expressions (ft their x)
(c) A1 length = 15 cm and width = 8 cm (both required, cao)
(c) Answer: length = 15 cm, width = 8 cm

Question 18

(a) M1 (2x + 20) + (3x + 10) + (x + 30) = 180 oe (angles on a straight line sum to 180 degrees)
(a) dM1 6x + 60 = 180 leading to 6x = 120 (dep)
(a) A1 x = 20 cao
(a) Answer: x = 20
(b) M1 substitutes x = 20 into each expression (ft their x)
(b) A1 60 degrees, 70 degrees and 50 degrees (all three correct)
(b) B1 correctly identifies 70 degrees, i.e. (3x + 10), as the largest angle
(b) Answer: 60 degrees, 70 degrees and 50 degrees; the largest angle is 70 degrees

Question 19

M1 Priya's age = x + 4 oe
M1 sum of ages = x + (x + 4) = 2x + 4, and mother's age = 3(2x + 4) - 8 oe
dM1 3(2x + 4) - 8 = 52 leading to 6x + 4 = 52 (dep on both previous marks)
A1 x = 8 cao, with the conclusion that Ben is 8 years old
Answer: x = 8; Ben is 8 years old

Question 20

(a) B1 Plan A cost = 15 + 2g oe
(a) B1 Plan B cost = 9 + 3.5g oe
(a) Answer: Plan A: 15 + 2g; Plan B: 9 + 3.5g
(b) M1 15 + 2g = 9 + 3.5g oe (ft their expressions from part a)
(b) dM1 6 = 1.5g oe (dep, collect terms correctly)
(b) A1 g = 4 cao
(b) Answer: g = 4
(c) B1 GBP 23 cao (ft their g)
(c) Answer: GBP 23

Question 21

M1 multiplies every term by 6, the LCM of 3 and 2, e.g. 2(2x - 1) + 3(x + 1) = 36 oe
dM1 expands and collects terms to reach 7x + 1 = 36 oe (dep)
A1 x = 5 cao
Answer: x = 5

Question 22

M1 cross multiplies correctly, 5(x - 4) = 2(2x + 1) oe
dM1 expands both sides and collects terms, 5x - 20 = 4x + 2 leading to x = 22 (dep)
A1 x = 22 cao
Answer: x = 22

Question 23

M1 expands the left-hand side correctly, 6x + 12 - 2x = 4x + 12 oe
M1 expands the right-hand side correctly, 4(x + 5) = 4x + 20 oe
B1 correct conclusion with valid reason (dep on both M marks), e.g. '4x + 12 = 4x + 20 gives 12 = 20, which is false for every value of x, so the equation has no solutions' oe
Answer: The equation has no solutions, since simplifying gives 12 = 20, which is never true.

Question 24

M1 expands the left-hand side correctly, 3x + 12 = kx + 12 oe
M1 recognises the constant terms already match (12 = 12), so the x-coefficients must also be equal for the equation to hold for every x (dep)
A1 k = 3 cao, with a correct supporting reason
Answer: k = 3

Solving Equations: Fluency and Exam Drill - Worksheets, Questions and Revision