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GCSE · Algebra

4.6D Forming and Solving Equations: Fluency and Exam Drill

EDEXCEL 1MA1 · Calculators not allowed · about 65 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Equation-Solving Toolkit

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To solve a linear equation, use inverse operations to get the unknown on its own, always doing the same thing to both sides. Expand any brackets first: a(b + c) = ab + ac. If the unknown appears on both sides, collect all the unknown terms on one side and all the number terms on the other before simplifying. For equations with fractions, multiply through by the denominator (or the lowest common denominator, if there is more than one) to clear the fractions before solving as usual. Work through the fluency questions first to build speed and accuracy with the balance method, then use the exam-style questions to practise forming your own equations from words and real-life contexts.

Solve x + 7 = 15

(Total for Question 1 is 1 mark)

Solve x - 9 = 4

(Total for Question 2 is 1 mark)

Solve 5x = 35

(Total for Question 3 is 1 mark)

Solve x/4 = 6

(Total for Question 4 is 1 mark)

Solve 3x + 4 = 19

(Total for Question 5 is 2 marks)

Solve 5x - 7 = 18

(Total for Question 6 is 2 marks)

Solve 2x + 5 = -3

(Total for Question 7 is 2 marks)

Solve 20 - 3x = 5

(Total for Question 8 is 2 marks)

Solve 4(x + 3) = 32

(Total for Question 9 is 2 marks)

Solve 3(2x - 5) = 21

(Total for Question 10 is 2 marks)

Solve 5x + 3 = 2x + 18

(Total for Question 11 is 2 marks)

Solve 7x - 4 = 3x + 20

(Total for Question 12 is 2 marks)

Solve x/3 + 2 = 9

(Total for Question 13 is 2 marks)

Solve (x + 1)/2 = 5

(Total for Question 14 is 2 marks)

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

(Total for Question 15 is 3 marks)

Three angles lie on a straight line. The angles are x degrees, (2x + 8) degrees and (3x - 2) degrees.

(Total for Question 16 is 3 marks)

Solve (x + 4)/3 = (2x - 1)/5

(Total for Question 17 is 4 marks)

The sum of three consecutive even numbers is 84. Let n be the smallest of the three numbers.

(a)Write an expression, in terms of n, for the sum of the three numbers.(1)

(b)Form and solve an equation to find the value of n.(3)

(c)Hence write down the three consecutive even numbers.(1)

(Total for Question 18 is 5 marks)

A triangle has angles of (2x + 10) degrees, (3x - 5) degrees and (x + 25) degrees.

(a)Show that 6x + 30 = 180(2)

(b)Hence find the size of the smallest angle in the triangle.(3)

(Total for Question 19 is 5 marks)

A carpenter charges a fixed call-out fee of 45 pounds plus 35 pounds per hour worked. Let C pounds be the total cost of a job that takes h hours.

(a)Write an expression for C in terms of h.(1)

(b)One job costs Sanjay 220 pounds in total. Form an equation and solve it to find how many hours the job took.(3)

(Total for Question 20 is 4 marks)

Amara is x years old. Her sister Bimpe is (x + 5) years old. The sum of their ages is 37. Form and solve an equation to find the value of x, then state whether Amara is a teenager (aged 13 to 19 inclusive), giving a reason.

(Total for Question 21 is 4 marks)

Solve (x + 2)/3 - (x - 1)/4 = 1

(Total for Question 22 is 4 marks)

Solve the equation 4(x + 3) - 2(x - 1) = 2(x + 7), and explain what your answer tells you about the equation.

(Total for Question 23 is 4 marks)

Mark scheme · 4.6D Forming and Solving Equations: Fluency and Exam Drill

Question 1

B1 x = 8 cao
Answer: x = 8

Question 2

B1 x = 13 cao
Answer: x = 13

Question 3

B1 x = 7 cao
Answer: x = 7

Question 4

B1 x = 24 cao
Answer: x = 24

Question 5

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

Question 6

M1 5x = 25 oe
A1 x = 5 cao
Answer: x = 5

Question 7

M1 2x = -8 oe
A1 x = -4 cao
Answer: x = -4

Question 8

M1 correct rearrangement, e.g. 15 = 3x oe, or -3x = -15 oe
A1 x = 5 cao
Answer: x = 5

Question 9

M1 correct method, e.g. expand to 4x + 12 = 32 oe, or divide first to x + 3 = 8 oe
A1 x = 5 cao
Answer: x = 5

Question 10

M1 correct method, e.g. expand to 6x - 15 = 21 oe, or divide first to 2x - 5 = 7 oe
A1 x = 6 cao
Answer: x = 6

Question 11

M1 correct collection of terms, 3x = 15 oe
A1 x = 5 cao
Answer: x = 5

Question 12

M1 correct collection of terms, 4x = 24 oe
A1 x = 6 cao
Answer: x = 6

Question 13

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

Question 14

M1 x + 1 = 10 oe
A1 x = 9 cao
Answer: x = 9

Question 15

M1 correct equation formed, 2(x+5) + 2(x-2) = 46 oe, or 4x + 6 = 46
M1 correct method to solve, 4x = 40 oe
A1 x = 10 cao
Answer: x = 10

Question 16

M1 correct equation formed, x + (2x+8) + (3x-2) = 180 oe
M1 correct simplification, 6x + 6 = 180 oe
A1 x = 29 cao
Answer: x = 29

Question 17

M1 correct cross-multiplication, 5(x+4) = 3(2x-1) oe
M1 correct expansion of both sides, 5x + 20 = 6x - 3 oe
dM1 dependent on both previous M marks, correct collection of terms, 23 = x oe
A1 x = 23 cao
Answer: x = 23

Question 18

(a) B1 3n + 6 oe (from n + (n+2) + (n+4))
(a) Answer: 3n + 6
(b) M1 3n + 6 = 84 oe (ft from part a)
(b) M1 3n = 78 oe
(b) A1 n = 26 cao
(b) Answer: n = 26
(c) B1 ft, 26, 28 and 30 all stated
(c) Answer: 26, 28, 30

Question 19

(a) M1 correct collection of the three angle expressions, (2x+10) + (3x-5) + (x+25) oe
(a) A1 cso, correctly simplified to 6x + 30 = 180, using angles in a triangle sum to 180 degrees, with all steps shown
(a) Answer: 6x + 30 = 180 (shown)
(b) M1 6x = 150 oe (ft from part a)
(b) M1 x = 25 (ft), substituted into all three angle expressions
(b) A1 50 degrees cao, correctly identified as the smallest angle (angles are 60, 70 and 50 degrees)
(b) Answer: 50 degrees

Question 20

(a) B1 C = 45 + 35h oe
(a) Answer: C = 45 + 35h
(b) M1 45 + 35h = 220 oe (ft from part a)
(b) M1 35h = 175 oe
(b) A1 h = 5 (hours) cao
(b) Answer: h = 5 hours

Question 21

M1 correct equation formed, x + (x+5) = 37 oe
M1 2x = 32 oe
A1 x = 16 cao
B1 ft, correct conclusion with reason, e.g. yes, Amara is a teenager because 13 ≤ 16 ≤ 19
Answer: x = 16; yes, Amara is a teenager since 13 ≤ 16 ≤ 19

Question 22

M1 correct common denominator used, multiplying every term by 12 to give 4(x+2) - 3(x-1) = 12 oe
M1 correct expansion, 4x + 8 - 3x + 3 = 12 oe
dM1 dependent on both previous M marks, correct simplification, x + 11 = 12 oe
A1 x = 1 cao
Answer: x = 1

Question 23

M1 LHS expanded correctly, 4x + 12 - 2x + 2 = 2x + 14 oe
M1 RHS expanded correctly, 2x + 14
A1 correctly identifies that both sides simplify to 2x + 14, so the equation is true for every value of x (an identity), not just one value
B1 valid supporting explanation, e.g. checking a numerical value of x (such as x = 0 or x = 5) and showing both sides give the same result
Answer: The equation is true for every value of x (an identity); there is no single solution.

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