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GCSE · Algebra

4.6 Forming and Solving Equations

EDEXCEL 1MA1 · Calculator allowed · about 85 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Solve each equation.

(a)x + 9 = 20(1)

(b)y - 8 = 10(1)

(Total for Question 1 is 2 marks)

Solve each equation. Show your working.

(a)4x + 5 = 25(2)

(b)6x - 7 = 29(2)

(Total for Question 2 is 4 marks)

Solve 4x + 9 = 1

(Total for Question 3 is 2 marks)

Solve 3(x + 4) = 27

(Total for Question 4 is 3 marks)

Solve 5x - 3 = 3x + 11

(Total for Question 5 is 2 marks)

Aisha thinks of a number. She multiplies the number by 3, then adds 7. The result is 31.

(Total for Question 6 is 3 marks)

A rectangle has one side of length (x + 3) cm and the adjacent side of length (2x - 1) cm. The perimeter of the rectangle is 34 cm.

Diagram NOT accurately drawn

(a)Form an equation in x and solve it.(3)

(b)Hence find the length of each side of the rectangle.(2)

(Total for Question 7 is 5 marks)

Three angles lie on a straight line. The angles are 2x degrees, 3x degrees and 40 degrees.

Diagram NOT accurately drawn

(a)Form an equation in x and solve it.(3)

(b)Work out the size of the two unknown angles.(2)

(Total for Question 8 is 5 marks)

Three consecutive integers have a sum of 96. Let n be the smallest integer.

(a)Form an equation in n and solve it.(2)

(b)Hence write down the three consecutive integers.(2)

(Total for Question 9 is 4 marks)

Solve x/3 + 5 = 12

(Total for Question 10 is 2 marks)

A taxi company charges a fixed fee of 3 pounds plus 2.50 pounds per mile travelled. A journey costs 20.50 pounds in total. Let m be the number of miles travelled.

(Total for Question 11 is 3 marks)

Solve 0.5x + 3.2 = 7.7

(Total for Question 12 is 2 marks)

Which value of x satisfies the equation 2x - 5 = 9 ?

A) x = 2
B) x = 5
C) x = 7
D) x = 9

(Total for Question 13 is 1 mark)

Show that the equation 4(2x - 3) = 2(x + 9) simplifies to x = 5

(Total for Question 14 is 2 marks)

A triangle is isosceles. The two equal base angles are each (2x + 5) degrees. The third angle is (x + 50) degrees.

Diagram NOT accurately drawn

(a)Form an equation in x and solve it.(3)

(b)Work out the size of the largest angle in the triangle, giving a reason for your answer.(2)

(Total for Question 15 is 5 marks)

Solve 4(x - 2) = 2(x + 5)

(Total for Question 16 is 3 marks)

Solve (2x - 1)/3 = 5

(Total for Question 17 is 3 marks)

Chloe is x years old. Her brother Daniel is (2x + 3) years old. In 6 years' time, the sum of their ages will be 45.

(a)Form an equation in x and show that it simplifies to 3x + 15 = 45(2)

(b)Solve the equation to find Chloe's age and Daniel's age now.(3)

(Total for Question 18 is 5 marks)

At a school fair, 3 raffle tickets and 2 programmes cost 5.10 pounds in total. 5 raffle tickets and 2 programmes cost 7.50 pounds in total. Let x pounds be the cost of one raffle ticket and y pounds be the cost of one programme.

(a)Write down a pair of simultaneous equations in x and y.(2)

(b)Solve your equations to find the cost of one raffle ticket and the cost of one programme.(3)

(Total for Question 19 is 5 marks)

A rectangle has length (x + 7) cm and width x cm. The area of the rectangle is 60 cm^2.

Diagram NOT accurately drawn

(a)Show that x² + 7x - 60 = 0(2)

(b)By factorising, solve the equation to find the width of the rectangle.(3)

(Total for Question 20 is 5 marks)

Mark scheme · 4.6 Forming and Solving Equations

Question 1

(a) B1 x = 11 cao
(a) Answer: x = 11
(b) B1 y = 18 cao
(b) Answer: y = 18

Question 2

(a) M1 4x = 20 oe
(a) A1 x = 5 cao
(a) Answer: x = 5
(b) M1 6x = 36 oe
(b) A1 x = 6 cao
(b) Answer: x = 6

Question 3

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

Question 4

M1 expand brackets correctly, 3x + 12 = 27 oe
M1 correct method to isolate x, 3x = 15 oe
A1 x = 5 cao
Answer: x = 5

Question 5

M1 correct method to collect x terms on one side, 2x - 3 = 11 oe
A1 x = 7 cao
Answer: x = 7

Question 6

B1 correct equation, 3x + 7 = 31 oe, where x is the number
M1 correct method to solve, 3x = 24 oe
A1 x = 8 cao
Answer: x = 8

Question 7

(a) B1 correct equation, 2(x + 3) + 2(2x - 1) = 34 oe, or 6x + 4 = 34
(a) M1 correct method to solve, 6x = 30 oe
(a) A1 x = 5 cao
(a) Answer: x = 5
(b) M1 substitute x = 5 (ft) into both expressions
(b) A1 8 cm and 9 cm cao, with units
(b) Answer: 8 cm and 9 cm

Question 8

(a) B1 correct equation, 2x + 3x + 40 = 180 oe, or 5x + 40 = 180
(a) M1 correct method to solve, 5x = 140 oe
(a) A1 x = 28 cao
(a) Answer: x = 28
(b) M1 substitute x = 28 (ft) into 2x and 3x
(b) A1 56 degrees and 84 degrees cao
(b) Answer: 56 degrees and 84 degrees

Question 9

(a) B1 correct equation, n + (n+1) + (n+2) = 96 oe, or 3n + 3 = 96
(a) A1 n = 31 cao
(a) Answer: n = 31
(b) M1 correct method, n+1 and n+2 found (ft)
(b) A1 31, 32 and 33 all cao
(b) Answer: 31, 32, 33

Question 10

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

Question 11

B1 correct equation, 3 + 2.5m = 20.5 oe
M1 correct method to solve, 2.5m = 17.5 oe
A1 m = 7 (miles) cao, unit required for full marks
Answer: m = 7 miles

Question 12

M1 0.5x = 4.5 oe
A1 x = 9 cao
Answer: x = 9

Question 13

B1 C, x = 7
Answer: C) x = 7

Question 14

M1 expand both sides correctly, 8x - 12 = 2x + 18 oe
A1 cso, correct rearrangement shown to give x = 5 with no errors
Answer: x = 5 (shown)

Question 15

(a) B1 correct equation, 2(2x+5) + (x+50) = 180 oe, or 5x + 60 = 180
(a) M1 correct method to solve, 5x = 120 oe
(a) A1 x = 24 cao
(a) Answer: x = 24
(b) M1 substitute x = 24 (ft) into x + 50 and into 2x + 5
(b) A1 74 degrees cao, with reason that 74 > 53
(b) Answer: 74 degrees

Question 16

M1 expand both sides correctly, 4x - 8 = 2x + 10 oe
M1 correct method to collect terms, 2x = 18 oe
A1 x = 9 cao
Answer: x = 9

Question 17

M1 multiply both sides by 3, 2x - 1 = 15 oe
M1 correct method to isolate x, 2x = 16 oe
A1 x = 8 cao
Answer: x = 8

Question 18

(a) M1 correct expressions for future ages, (x+6) + (2x+3+6) = 45 oe
(a) A1 cso, correctly simplified to 3x + 15 = 45 with all steps shown
(a) Answer: 3x + 15 = 45 (shown)
(b) M1 3x = 30 oe (ft from part a)
(b) A1 x = 10 (Chloe's age) cao
(b) B1 ft, Daniel's age = 23 (2*10 + 3)
(b) Answer: Chloe is 10, Daniel is 23

Question 19

(a) B1 3x + 2y = 5.10 oe
(a) B1 5x + 2y = 7.50 oe
(a) Answer: 3x + 2y = 5.10 and 5x + 2y = 7.50
(b) M1 correct elimination method, subtract equations to get 2x = 2.40 oe
(b) A1 x = 1.20 (ft)
(b) A1 y = 0.75 (ft)
(b) Answer: Raffle ticket = 1.20 pounds, programme = 0.75 pounds

Question 20

(a) M1 correct area expression, x(x+7) = 60 oe
(a) A1 cso, correctly expanded and rearranged to x² + 7x - 60 = 0 with all steps shown
(a) Answer: x² + 7x - 60 = 0 (shown)
(b) M1 correct factorisation, (x+12)(x-5) = 0 oe, or valid alternative method
(b) A1 x = -12 or x = 5 both found
(b) A1 x = 5 cm selected (dep), with reason that width must be positive
(b) Answer: x = 5 cm

Forming and Solving Equations - Worksheets, Questions and Revision

4.6 Forming and Solving Equations

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Question 19

Question 20