Worksheets for GCSE Mathematics. Solving Equations. Mr Black's Maths Resources for Teachers GCSE 1-9. Algebra

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1 Worksheets for GCSE Mathematics Solving Equations Mr Black's Maths Resources for Teachers GCSE 1-9 Algebra

2 Equations Worksheets Contents Differentiated Independent Learning Worksheets Solving Equations using Function Machines Solving Equations with the Balance Method Equations with the Unknown on Both Sides Equations with Fractions Trial and Improvement Solving Simultaneous Equations Graphically Simultaneous Equations by Elimination Problems Involving Simultaneous Equations Page 30 Page 40 Page 0 Page 60 Page 70 Page 80 Page 90 Page 100 Solutions Solving Equations using Function Machines Solving Equations with the Balance Method Equations with the Unknown on Both Sides Equations with Fractions Trial and Improvement Solving Simultaneous Equations Graphically Simultaneous Equations by Elimination Problems Involving Simultaneous Equations Page 11 Page 12 Page 13 Page 14 Page 1 Page 16 Page 17 Page 18 2

3 Q1. Use function machines to solve the following equations. a) xx + 7 = 13 b) xx 8 = c) 8xx = 48 8 d) xx 6 = 4 7 Q2. Use function machines to solve the following equations. a) 3xx + 4 = 31 b) 6xx 7 = c) xx 4 + = 11 d) xx 7 1 = e) 2(xx 3) = 8 f) 4( + xx) = 48 Use function machines to solve the following equations. a) 4xx + 6 = 30 b) xx 4 = 11 c) 12xx 9 = 1 d) xx + 3 = 10 8 e)6xx 7 = 3 f) xx + 8 = 11 3 g) xx 6 = 0 6 h) xx + 3 = 12 9 i) 2(xx + 4) = 22 j) 3(xx + ) = 21 k) xx 10 = 1 6 l) (3 + xx) = 60 3

4 Q1. Solve these equations using the balance method. a) 3xx + 7 = 19 b) xx 4 = 21 c)6xx 7 = 41 d) 2xx 8 = 16 e7xx 10 = 3 f) xx + = 9 g) xx + 1. = 4 4 h) 4xx + = 11 i) xx 3 = 0 2 Q2. Solve these equations using the balance method. a) 2(xx 1) = 4 b)4(xx + 3) = 32 c) 6(xx 4) + 3 = 3 d) 1 2 (2xx + 6) = 9 e)3(2 + xx) = 33 f) 2xx+1 = g) 4xx = 1 h) 6+2xx 10 3xx+7 = 2 i) = 8 Match the solutions with the equations. 3(xx 1) = 9 + 3xx = 14 4(xx 1.) = 8 4xx 3 = 7 xx = 3 xx = 3. xx = 2. xx = 4 Q4. a) A rectangle has a length that is three cm greater than its width. Its perimeter is 26 cm. xx cccc Calculate the length and width of the rectangle. xx + 3 cccc b) This isosceles triangle has a total perimeter of 72 units. Set up and solve an equation to calculate the length of each side. 2xx 3xx + 4 3xx + 4 c) Luke thinks of a number. When this number is multiplied by and added to 12 the result is 32. What number did Luke think of? d) Mary is 6 years older than Jane. Mary and Jane have a total age of 2 years. How old is Mary? e) The sum of three consecutive integers is 24. What are the three integers? 4

5 Q1. Solve the following equations: a) cc + 6 = 6cc + 1 b) yy + 17 = yy 3 c) u = 12 - u d) 3ff + 12 = ff + 4 e) nn + 2 = 3nn + 8 f) 4g + 1 = g + 7 g) 3 + i = 2i + 12 h) 2kk + 11 = 6kk + 3 i) 3r 18 = 9r + 3 j) 2yy + 7 = yy k) 8u + 3 = u + 2 l) 6p 2 = 4p + m) 3mm + 16 = 4mm + n) 10a + 4 = a o) 6d 1 = d + 14 Q2. a) 6(yy + 3) = 2yy + 22 b) 10(1 + tt) = 3(tt + 8) c) (zz 3) = zz + 17 d) 3(jj + 4) = 4(jj + 1) e) 7(xx 1) = 3(xx + 3) f) 4(uu + 1) = 19 uu g) 3(nn 2) = 2(4 2nn) h) 3(4 + vv) = (vv 4) i) 2qq + 16 = (qq 7) j) 2(1 ee) = 3ee 8 k) (10 2aa) = aa l) 3( xx) = 3xx 1 a) Matthew thinks of a number, multiplies it by three and subtracts two. He get the same result when he multiplies the original number by two and adds. What number was Matthew thinking of? b) Martin doubles a number and adds three to the result. Ian adds five to his number. Martin and Ian get the same result. What number did they each think of? c) These two rectangles have the same perimeter. Calculate the value of xx. Q. a) Explain why the equation 2(xx + ) = 2xx 4 can never be solved. b) Explain why there are an infinite number of solutions to the equation 3(4xx 2) = 2(6xx 3).

6 Q1. Solve the following equations. Equations with Fractions a) x + 2 = 8 b) 3m 2 1 = 8 c) 2y + = 12 d) 6x+3 9 g) 6(x+1) = 1 e) r+3 4 = 9 h) (3 2x) 8 = 1 f) 2e 3 = 11 i) 3(1 4x) 8 = 4 = 4 Q2. Solve these equations. a) 2x+3 4 d) 2(3x+1) 7 = 2x + 1 b) 6+2g = 2 x e) (3 r) 4 = 3g + 1 c) 4r 3 = 3 + 2r f) 2( 7w) 3 = 2r 3 = w + 1 Solve these equations. a) 2 3x 4 = 1+2x 3 b) 3(x 2) 2 2 = x 4 c) 3f = f d) y+3 = 2 + y 3 4 e) 16 c 8 2(4c+10) = 2 f) 2 1 n = n g) 3(t+6) t = 2 h) 6 2w = 2+w i) 3y 4 4 y+1 3 = 1 y 3 Q4. Match the equation with its correct solution. 2m + m 3 = 21 m = 1 m 1 = m 3 2 m = m 2 = + m 3 m = 3 m 2 m 3 2(1 m) = 4 3 m = 6

7 Q1. Solve these equations using a method of trial and improvement to one decimal place. a) xx 2 + xx = 2 xx xx 2 + xx High/Low 4 20 Low 0 High b) xx 2 2xx = 4 xx xx 2 2xx High/Low 3 3 Low 4 8 High c) xx 2 + 3xx = 147 xx xx 2 + 3xx High/Low Low High d) xx 2 4xx + = 210 xx xx 2 4xx + High/Low Low High Q2. Solve these equations using a method of trial and improvement to one decimal place. a) xx 3 xx = 100 b) xx 3 + 2xx = 40 xx xx 3 xx High/Low 4 60 Low 120 High xx xx 3 + 2xx High/Low 3 33 Low 4 72 High c) xx 3 + xx 2 = 20 xx xx 3 + xx 2 High/Low 2 12 Low 3 36 High d) xx 3 + xx 2 + 4xx = 1000 xx xx 3 + xx 2 + 4xx High/Low Low High a) Use a method of trial and improvement to solve the equation xx 3 + xx = 8 correct to one decimal places. b) Use a method of trial and improvement to solve the equation 2xx 2 1 = 9 correct to two xx decimal places. c) Use a method of trial and improvement to solve the equation xx2 place. xx = 20 correct to one decimal d) Use a method of trial and improvement to solve the equation xx3 2 = 10 correct to two decimal places. xx 7

8 Q1. Draw graphs to solve these pairs of equations. a) xx + yy = 2xx + 4yy = 12 b) xx + yy = 7 xx + 3yy = 12 Q2. Solve these pairs of simultaneous equations using graphical methods. 2xx + yy = 10 & xx + 2yy = 14 Simon and Donna go shopping. Simon buys three DVDs and two Blue-Rays for 24. Donna buys one DVD and four Blue-Rays for 28. a) b) c) Set up two equations using these facts. Plot the two equations on a pair of axes. Find the cost of a DVD and Blue-Ray, Q4 Ellie and Archie bought some books. Ellie buys one soft bound book and two hard bounds for a cost of 16. Archie buys 3 soft bound books and two hard bound books for 24. Use graphical methods to find the cost of 4 soft bound and 3 hard bound books. 8

9 Q1. Solve the following pairs of equations simultaneously. a) 2xx + yy = 4 b) 3xx + 2yy = 12 c) xx + 2yy = 11 xx + yy = 3 xx + 2yy = 8 xx + 3yy = 1 d) 2xx yy = 7 e) xx 4yy = 3 f) 3xx 2yy = 6 xx yy = 2 2xx + 4yy = 18 2xx 2yy = 2 g) 4xx + 3yy = 13 h) xx + yy = 14 i) xx + 3yy = 4 2xx 3yy = 11 3xx + yy = 10 xx yy = 0 j) 3xx + 2yy = 28 k) 6xx 2yy = 33 l) xx + 3yy = 6 3xx + 3yy = 33 6xx 6yy = 12 xx yy = 8 Q2. a) Two numbers a and b have a sum of 86 and a difference of 16. Set up a pair of equations to calculate the values of a and b. b) The sum of ages of Harry and Matilda is 88 years. The difference of their ages is 6 years. If Harry is older than Matilda what are their ages? c) Aidan pays 7.00 for two pens and three pencils. Brandon pays for two pens and one pencil. What is the price for a pen and pencil? d) A coach journey costs each adult a and each child c. Tickets for one adult and two children cost 12. Tickets for two adults and two children cost Calculate the cost of each type of ticket. Set up a pair of simultaneous equations to calculate the length of the third diagram. 22 cm 17 cm? cm 9

10 Q1. Solve the following pairs of equations simultaneously. a) xx 7yy = 23 3xx + 9yy = 9 d) 4xx yy = 2xx 3yy = 2 g) 3xx + 2yy = 8 2xx yy = 3 j) 8xx + 3yy = 2 xx + 2yy = 1 b) 2xx 2yy = 8 7xx yy = 12 e) 6xx 2yy = 9 3xx + 4yy = 12 h) xx + yy = xx 3yy = 1 k) 2xx + 3yy = 14 8xx yy = c) 6xx 2yy = 9 3xx + 4yy = 12 f) xx + 10yy = 4 2xx 8yy = 0 i) 3xx 2yy = 4 xx + 4yy = 6 l) 3xx + 7yy = 2 4xx + 3yy = 9 Q2. Arrange each pair equations so the unknowns are aligned and solve simultaneously. a) yy = 3xx 7 3xx + 2yy = 13 d) yy 10 = 2xx 2xx 3yy = 4 b) xx = 1 + yy 7xx yy = 17 e) 2xx + yy = 18 3xx + 7 = yy c) 1 + 3xx = yy 3xx + 4yy = 34 f) 7yy 4xx = 2 0 = 4xx 6yy a) Luke pays 2.2 for three Blu-ray movies and a video console game. Graham pays 71 for two Blu-ray movies and three video console games. What is the price for a single Blue-ray and video console game? b) At the local green grocery the price of five apples and four oranges is The price for one apple and three oranges is What is the individual price for an apple and orange? c) Two small jars and one large jar can hold 8 cups of water. Three large jars minus one small jar constitute 10 cups of water. How many cups of water can each jar hold? Q4. In the diagram to the right the sum of each column and row is shown. Use this information to calculate the value of each symbol. 10

11 Solutions Q1. a) xx = 6 b) xx = 13 c) xx = 6 d) xx = 24 Q2 a) xx = 9 b) xx = c) xx = 24 d) xx = 42 e) xx = 7 f) xx = 7 a) xx = 6 b) xx = 3 c) xx = 2 d) xx = 6 e)xx = 7 f) xx = 9 g) xx = 36 h) xx = 81 i) xx = 7 j) xx = 2 k)xx = 66 l) xx = 9 11

12 Solutions Q1. a) xx = 4 b) xx = c) xx = 8 d) xx = 12 e) xx = 9 f) xx = 20 g) xx = 10 h) xx = 1. i) xx = 6 Q2. a) xx = 3 b) xx = c) xx = 4 d) xx = 6 e) xx = 9 f) xx = 12 g) xx = 2. h) xx = 7 i) xx = 11 3(xx 1) = 9 + 3xx = 14 4(xx 1.) = 8 4xx 3 = 7 xx = 3 xx = 3. xx = 2. xx = 4 Q4. a) Length = 8 cm, Width = cm b) xx = 8, Sides = 16, 28 and 28. c) Luke s number = 4 d) Mary = 23 years old e) Integers are 7, 8 and 9 12

13 Q1. a) cc = 1 b) yy = c) u = 2 d) ff = 4 e) nn = 3 f) g = 2 g) ii = 3 h) kk = 2 i) r = 3. j) yy = 4 k) uu = 7 l) p = 3. m) mm = 11 n) a = 6 o) d = 3 Q2. a) yy = 1 b) tt = 2 c) z = 8 d) j = 8 e) xx = 4 f) u = 3 g) n = 2 h) v = 16 i) q = 17 j) e = 2 k) a = l) xx = a) 3xx 2 = 2xx + xx = 7 b) 2xx + 3 = xx + xx = 2 c) 6xx + 6 = 8xx + 4 xx = 1 Q. a) 0 14 oooo 14 b) Each side is an equivalent expression. 13

14 Solutions Equations with Fractions Q1. a) x = 30 b) m = 6 c) y = 17. d) x = 1 e) r = 0.2 f) e = 11. g) x = 6. h) x = 7.3 i) x = Q2. a) x = 1 6 b) g = 1 13 c) r = 2 d) x = e) r = 3 13 f) w = 7 29 a) x = 2 17 b) x = 3 c) f = 9 2 d) y = 13 e) c = 0 f) n = 13 2 g) ) t = 8 h) w = 3 i) y = 20 7 Q4. 2m + m 3 = 21 m = 1 m 1 = m 3 2 m = m 2 = + m 3 m = 3 m 2 m 3 2(1 m) = 4 3 m = 14

15 Solutions Q1. a) xx = 4. tttt 1 dddd b) xx = 3.2 tttt 1 dddd c) xx = 10.7 tttt 1 dddd d) xx = 16. tttt 1dddd Q2. a) xx = 4.7 tttt 1 dddd b) xx = 3.2 tttt 1 dddd c) xx = 2.4 tttt 1 dddd d) xx = 9. tttt 1dddd a) xx = 1.8 tttt 1 dddd b) xx = 2.17 tttt 2dddd or xx = 2.06 tttt 2dddd or xx = 0.11 tttt 2dddd c) xx = 4.1 tttt 1 dddd or xx = 24.1 tttt 1 dddd d) xx = 12.2 tttt 2 dddd or xx = 0.01 tttt 2 dddd or xx = tttt 2 dddd 1

16 Solutions Q1. a) xx = 4, yy = 1 b) xx = 4., yy = 2. Q2. xx = 2, yy = 6 DVD = 4, Blue-Ray = 6 Q4. 34 Soft Bound = 4 Hard Bound = 6 16

17 Q1. a) xx = 1, yy = 2 b) xx = 2, yy = 3 c) xx = 3, yy = 4 d) xx =, yy = 3 e) xx = 7, yy = 1 f) xx = 4, yy = 3 g) xx = 4, yy = 1 h) xx = 2, yy = 4 i) xx = 1, yy = 1 j) xx = 6, yy = k) xx = 7.2, yy =.2 l) xx = 0.9, yy = 3. Q2. a) xx = 1, yy = 3 b) Harry is 47, Matilda is 41 c) Pen = 2, Pencil = 1 d) Adult =.0, Child = 3.2 3xx + yy = 22, 2xx + yy = 17 Width = 7, Length = Total length of third shape = 36 cm 17

18 Solutions Q1. a) xx = 9, yy = 2 b) xx = 4, yy = 8 c) xx = 2, yy = 1. d) xx = 2., yy = 1 e) xx = 2, yy = 1. f) xx = 6, yy = 1. g) xx = 2, yy = 1 h) xx = 2, yy = 3 i) xx = 2, yy = 1 j) xx = 1, yy = 2 k) xx = 2., yy = 3 l) xx = 3, yy = 1 Q2. a) d) xx = 3, yy = 2 b) xx = 6, yy = c) xx = 2, yy = 7 xx = 2., yy = 3 e) xx = 1, yy = 4 f) xx = 3, yy = 2 a) Blu-ray = 12.2, Console Game = 1.0 b) Apple = 72p, Orange = 64p c) Large Jar = 4 cups, Small Jar = 2 cups Q4. =4 =-2 = 18