Solving Equations - Worksheets, Questions and Revision

18 original exam-style questions - 9 pages of questions with a full mark scheme - free printable PDF.

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GCSE · Mathematics

3.14 Solving Equations

EDEXCEL 1MA1 · Calculator allowed · about 95 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Revision notes: Solving Equations

Solving an equation means finding the value of the letter that makes it true. Beyond one-step equations, GCSE questions include two-step equations, equations with brackets, and equations with the letter on both sides, each needing operations applied to both sides in the correct order to keep the equation balanced.

Expand any brackets first, multiplying every term inside by the number (or letter) outside.
If the letter appears on both sides, add or subtract a letter term from both sides so it appears on only one side.
Collect number terms on the other side by adding or subtracting from both sides.
Finally divide (or multiply) both sides by the coefficient of the letter to leave the letter on its own, then check by substituting back in.

Worked example. Solve 3(x + 4) = 2x + 19.

Expand the bracket: 3x + 12 = 2x + 19.
Subtract 2x from both sides (letter on both sides): 3x - 2x + 12 = 19, so x + 12 = 19.
Subtract 12 from both sides: x = 19 - 12 = 7.
Check: 3(7 + 4) = 3 x 11 = 33; 2(7) + 19 = 14 + 19 = 33. Both sides match.

Answer: x = 7.

Solve each equation.

(a)x + 5 = 12(1)

(b)y - 7 = 3(1)

(d)x / 3 = 6(1)

(Total for Question 1 is 4 marks)

Solve each equation. Show your working.

(a)2x + 3 = 11(2)

(b)3x - 4 = 11(2)

(d)4x - 7 = 1(2)

(Total for Question 2 is 8 marks)

Solve each equation. Show your working.

(a)x / 2 + 3 = 7(2)

(b)(x + 4) / 3 = 5(2)

(c)-2x + 5 = 13(2)

(Total for Question 3 is 6 marks)

Solve each equation.

(a)0.5x + 3 = 7.5(2)

(b)2.5x - 4 = 6(2)

(Total for Question 4 is 4 marks)

Solve each equation. Show your working.

(a)3(x + 2) = 21(2)

(b)5(2x - 1) = 35(3)

(Total for Question 5 is 8 marks)

Solve each equation. Show your working.

(a)5x + 3 = 2x + 18(3)

(b)7x - 4 = 3x + 20(3)

(Total for Question 6 is 9 marks)

Solve each equation. Show your working.

(a)3(x + 4) = 2(x + 9)(3)

(b)4(2x - 3) = 3(x + 5)(3)

(Total for Question 7 is 6 marks)

Jess solves the equation 4x - 5 = 11. Her working is shown below.
4x = 11 - 5
4x = 6
x = 1.5
Jess's answer is wrong.

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

(b)Work out the correct value of x.(2)

(Total for Question 8 is 3 marks)

The three angles of a triangle are 3x degrees, (2x + 10) degrees and (x + 20) degrees.
Work out the value of x.

Diagram NOT accurately drawn

(Total for Question 9 is 3 marks)

A rectangle has length (3x - 2) cm and width (x + 4) cm. The perimeter of the rectangle is 60 cm.

Diagram NOT accurately drawn

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

(b)Work out the area of the rectangle.(2)

(Total for Question 10 is 5 marks)

Priya thinks of a number. She multiplies it by 4, then adds 7. The result is 39.
Using x for Priya's number, form an equation and solve it to find the value of x.

(Total for Question 11 is 3 marks)

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

(Total for Question 12 is 3 marks)

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

(Total for Question 13 is 3 marks)

Solve each equation. Give both values of x.

(a)x² = 49(2)

(b)x² - 5 = 20(2)

(Total for Question 14 is 4 marks)

Solve each quadratic equation by factorising.

(a)x² + 5x + 6 = 0(3)

(b)x² - 2x - 15 = 0(3)

(Total for Question 15 is 6 marks)

Solve the simultaneous equations
x + y = 10
x - y = 4
Show clear algebraic working.

(Total for Question 16 is 4 marks)

Which value of x satisfies the equation 3x - 2 = 13 ?
A) 3 B) 5 C) 15 D) 11/3

A) 3
B) 5
C) 15
D) 11/3

(Total for Question 17 is 1 mark)

Sam and Priya share out some money. Sam receives (3x + 5) pounds. Priya receives (x + 15) pounds. They both receive the same amount of money.

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

(b)Work out how much money Sam receives.(2)

(Total for Question 18 is 5 marks)

Mark scheme · 3.14 Solving Equations

Question 1

(a) B1 x = 7 cao
(a) Answer: x = 7
(b) B1 y = 10 cao
(b) Answer: y = 10
(c) B1 x = 5 cao
(c) Answer: x = 5
(d) B1 x = 18 cao
(d) Answer: x = 18

Question 2

(a) M1 2x = 8 oe
(a) A1 x = 4 cao
(a) Answer: x = 4
(b) M1 3x = 15 oe
(b) A1 x = 5 cao
(b) Answer: x = 5
(c) M1 5x = -15 oe
(c) A1 x = -3 cao
(c) Answer: x = -3
(d) M1 4x = 8 oe
(d) A1 x = 2 cao
(d) Answer: x = 2

Question 3

(a) M1 x / 2 = 4 oe
(a) A1 x = 8 cao
(a) Answer: x = 8
(b) M1 x + 4 = 15 oe
(b) A1 x = 11 cao
(b) Answer: x = 11
(c) M1 -2x = 8 oe
(c) A1 x = -4 cao
(c) Answer: x = -4

Question 4

(a) M1 0.5x = 4.5 oe
(a) A1 x = 9 cao
(a) Answer: x = 9
(b) M1 2.5x = 10 oe
(b) A1 x = 4 cao
(b) Answer: x = 4

Question 5

(a) M1 3x + 6 = 21 oe
(a) A1 x = 5 cao
(a) Answer: x = 5
(b) M1 10x - 5 = 35 (expand correctly)
(b) M1 10x = 40 (dep on first M1)
(b) A1 x = 4 cao
(b) Answer: x = 4
(c) M1 2x + 10 - 3 = 17 (expand correctly)
(c) M1 2x = 10 (dep on first M1)
(c) A1 x = 5 cao
(c) Answer: x = 5

Question 6

(a) M1 3x + 3 = 18 or 3x = 15 (collect x terms correctly)
(a) M1 3x = 15 (dep, rearrange constants)
(a) A1 x = 5 cao
(a) Answer: x = 5
(b) M1 4x - 4 = 20 or 4x = 24 (collect x terms correctly)
(b) M1 4x = 24 (dep, rearrange constants)
(b) A1 x = 6 cao
(b) Answer: x = 6
(c) M1 5x + 9 = 20 or 5x = 11 (collect x terms correctly)
(c) M1 5x = 11 (dep, rearrange constants)
(c) A1 x = 2.2 oe (11/5)
(c) Answer: x = 2.2

Question 7

(a) M1 3x + 12 = 2x + 18 (expand both brackets correctly)
(a) M1 x = 6 (dep, collect terms)
(a) A1 x = 6 cao
(a) Answer: x = 6
(b) M1 8x - 12 = 3x + 15 (expand both brackets correctly)
(b) M1 5x = 27 (dep, collect terms)
(b) A1 x = 5.4 oe (27/5)
(b) Answer: x = 5.4

Question 8

(a) B1 states she subtracted 5 instead of adding 5 to both sides oe
(a) Answer: She should have added 5 to both sides, not subtracted 5.
(b) M1 4x = 16 oe
(b) A1 x = 4 cao
(b) Answer: x = 4

Question 9

P1 3x + (2x + 10) + (x + 20) = 180 oe (angles in a triangle sum to 180 degrees)
M1 6x + 30 = 180 leading to 6x = 150 (dep)
A1 x = 25 cao
Answer: x = 25

Question 10

(a) P1 2[(3x - 2) + (x + 4)] = 60 oe (correct perimeter equation)
(a) M1 8x + 4 = 60 leading to 8x = 56 (dep)
(a) A1 x = 7 cao
(a) Answer: x = 7
(b) M1 length = 19 and width = 11 (ft their x, or correct substitution)
(b) A1 209 cm² cao
(b) Answer: 209 cm²

Question 11

P1 4x + 7 = 39 oe
M1 4x = 32 (dep)
A1 x = 8 cao
Answer: x = 8

Question 12

M1 2(2x + 1) = 3(x + 5) (correct cross multiplication)
M1 4x + 2 = 3x + 15 leading to x = 13 (dep)
A1 x = 13 cao
Answer: x = 13

Question 13

M1 3(x - 2) = 24 (multiply both sides by 4)
M1 x - 2 = 8 (dep, expand or divide by 3)
A1 x = 10 cao
Answer: x = 10

Question 14

(a) M1 square root process shown, e.g. x = √49
(a) A1 x = 7 and x = -7 (both values, oe ±7)
(a) Answer: x = 7 or x = -7
(b) M1 x² = 25 oe
(b) A1 x = 5 and x = -5 (both values, oe ±5)
(b) Answer: x = 5 or x = -5

Question 15

(a) M1 (x + 2)(x + 3) = 0 (correct factorisation)
(a) A1 x = -2
(a) A1 x = -3
(a) Answer: x = -2 or x = -3
(b) M1 (x - 5)(x + 3) = 0 (correct factorisation)
(b) A1 x = 5
(b) A1 x = -3
(b) Answer: x = 5 or x = -3

Question 16

M1 adds the two equations to eliminate y, e.g. 2x = 14
A1 x = 7
M1 substitutes x = 7 into either equation (dep)
A1 y = 3
Answer: x = 7, y = 3

Question 17

B1 B cao
Answer: B) x = 5

Question 18

(a) P1 3x + 5 = x + 15 oe
(a) M1 2x = 10 (dep)
(a) A1 x = 5 cao
(a) Answer: x = 5
(b) M1 substitutes x = 5 into 3x + 5 (ft their x)
(b) A1 GBP 20 cao
(b) Answer: GBP 20

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Solving Equations - Worksheets, Questions and Revision

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3.14 Solving Equations

Revision notes: 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 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