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Use a suitable method to solve the following pairs of simultaneous equations.

1. Use a suitable method to solve the following pairs of simultaneous equations. ( i ). x + 2 y = 20. 9 x − 2 y = 80. x + 2 y = 20. 9 x − 2y = 80 (Add rows). 10 x = 100 (Divide both sides by 10). x = 10. Let x = 10: x + 2 y = 20.

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Use a suitable method to solve the following pairs of simultaneous equations.

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  1. 1. Use a suitable method to solve the following pairs of simultaneous equations. (i) x + 2y = 20 9x − 2y = 80 x + 2y = 20 9x − 2y = 80 (Add rows) 10x = 100 (Divide both sides by 10) x = 10 Let x = 10: x + 2y = 20 10 + 2y = 20 (Subtract 10 from both sides) 2y = 10 (Divide both sides by 2) y = 5

  2. 1. Use a suitable method to solve the following pairs of simultaneous equations. (ii) 5p − 3q = −3 7p + 3q = 39 5p – 3q = −3 7p + 3q = 39 (Add rows) 12p = 36 (Divide both sides by 12) p = 3

  3. 1. Use a suitable method to solve the following pairs of simultaneous equations. (ii) 5p − 3q = −3 7p + 3q = 39 Let p = 3: 5p – 3q = − 3 5(3) – 3q = − 3 15 – 3q = − 3 (Add 3q to both sides) 15 = – 3 + 3q (Add 3 to both sides) 18 = 3q (Divide both sides by 3) 6 = q

  4. 1. Use a suitable method to solve the following pairs of simultaneous equations. (iii) 2a + 5b = 26 5a + 5b = 35 2a + 5b = 26 (Multiply both sides by − 1) 5a + 5b = 35 –2a – 5b = – 26 5a + 5b = 35 (Add rows) 3a = 9 (Divide both sides by 3) a = 3

  5. 1. Use a suitable method to solve the following pairs of simultaneous equations. (iii) 2a + 5b = 26 5a + 5b = 35 Let a = 3: 2a + 5b = 26 2(3) + 5b = 26 6 + 5b = 26 (Subtract 6 from both sides) 5b = 20 (Divide both sides by 5) b = 4

  6. 1. Use a suitable method to solve the following pairs of simultaneous equations. (iv) 7x − 2y = 22 5x − 2y = 14 7x – 2y = 22 5x – 2y = 14 (Multiply both sides by − 1) 7x – 2y = 22 –5x + 2y = – 14 (Add rows) 2x = 8 (Divide both sides by 2) x = 4

  7. 1. Use a suitable method to solve the following pairs of simultaneous equations. (iv) 7x − 2y = 22 5x − 2y = 14 Let x = 4: 5x − 2y = 14 5(4) – 2y = 14 20 – 2y = 14 (Add 2y to both sides) 20 = 14 + 2y (Subtract 14 from both sides) 6 = 2y (Divide both sides by 2) 3 = y

  8. 1. Use a suitable method to solve the following pairs of simultaneous equations. (v) 3m + 2n = 26 4m − n = 20 3m + 2n = 26 4m − n = 20 (Multiply both sides by 2) 3m + 2n = 26 8m – 2n = 40 (Add rows) 11m = 66 (Divide both sides by 11) m = 6

  9. 1. Use a suitable method to solve the following pairs of simultaneous equations. (v) 3m + 2n = 26 4m − n = 20 Let m = 6: 3m + 2n = 26 3(6) + 2n = 26 18 + 2n = 26 (Subtract 18 from both sides) 2n = 8 (Divide both sides by 2) n = 4

  10. 1. Use a suitable method to solve the following pairs of simultaneous equations. (vi) 4a + 3b = 20 3a − 4b = −10 4a + 3b = 20 (Multiply both sides by 4) 3a − 4b = −10 (Multiply both sides by 3) 16a + 12b = 80 9a – 12b = –30 (Add rows) 25a = 50 (Divide both sides by 25) a = 2

  11. 1. Use a suitable method to solve the following pairs of simultaneous equations. (vi) 4a + 3b = 20 3a − 4b = −10 Let a = 2: 4a + 3b = 20 4(2) + 3b = 20 8 + 3b = 20 (Subtract 8 from both sides) 3b = 12 (Divide both sides by 3) b = 4

  12. 1. Use a suitable method to solve the following pairs of simultaneous equations. (vii) 5x − 3y = 14 9x − 4y = 28 5x – 3y = 14 (Multiply both sides by − 4) 9x − 4y = 28 (Multiply both sides by 3) –20x + 12y = –56 27x – 12y = 84 (Add the rows) 7x = 28 (Divide both sides by 7) x = 4

  13. 1. Use a suitable method to solve the following pairs of simultaneous equations. (vii) 5x − 3y = 14 9x − 4y = 28 Let x = 4: 5x − 3y = 14 5(4) – 3y = 14 20 – 3y = 14 (Add 3y to both sides) 20 = 14 + 3y (Subtract 14 from both sides) 6 = 3y (Divide both sides by 3) 2 = y

  14. 1. Use a suitable method to solve the following pairs of simultaneous equations. (viii) 2x + 9y = 69 x + 3y = 24 2x + 9y = 69 x + 3y = 24 (Multiply both sides by −3) 2x + 9y = 69 −3x – 9y = −72 (Add rows) – x = – 3 (Multiply both sides by −1) x = 3

  15. 1. Use a suitable method to solve the following pairs of simultaneous equations. (viii) 2x + 9y = 69 x + 3y = 24 Let x = 3: x + 3y = 24 3 + 3y = 24 (Subtract 3 from both sides) 3y = 21 (Divide both sides by 3) y = 7

  16. 1. Use a suitable method to solve the following pairs of simultaneous equations. (ix) 6p + 2q = 14 5p + 5q = 5 6p + 2q = 14 (Multiply both sides by 5) 5p + 5q = 5 (Multiply both sides by − 2) 30p + 10q = 70 −10p – 10q = −10 (Add rows) 20p = 60 (Divide both sides by 20) p = 3

  17. 1. Use a suitable method to solve the following pairs of simultaneous equations. (ix) 6p + 2q = 14 5p + 5q = 5 Let p = 3: 6p + 2q = 14 6(3) + 2q = 14 18 + 2q = 14 (Subtract 18 from both sides) 2q = − 4 (Divide both sides by 2) q = − 2

  18. 2. Use a suitable method to solve the following simultaneous equations.In some cases it may be necessary to write the equation in the form ax + by = c, before solving. (i) 4x = 7 + 3y y + 3 = x Use the substitution method: From the second equation: y + 3 = x Substitute y + 3 in for x in the first equation: 4x = 7 + 3y 4x – 3y = 7 4(y + 3) – 3y = 7 4y + 12 – 3y = 7 (Subtract 12 from both sides) y = – 5

  19. 2. Use a suitable method to solve the following simultaneous equations.In some cases it may be necessary to write the equation in the form ax + by = c, before solving. (i) 4x = 7 + 3y y + 3 = x Let y = – 5: y + 3 = x – 5+ 3 = x – 2 = x

  20. 2. Use a suitable method to solve the following simultaneous equations.In some cases it may be necessary to write the equation in the form ax + by = c, before solving. (ii) 6x = 52 − 2y 70 − 7y = 5x 6x = 52 – 2y  6x + 2y = 52 (Multiply both sides by 7) 70 – 7y = 5x5x + 7y = 70 (Multiply both sides by − 2) 42x + 14y = 364 −10x − 14y = −140 32x = 224 (Divide both sides by 32) x = 7

  21. 2. Use a suitable method to solve the following simultaneous equations.In some cases it may be necessary to write the equation in the form ax + by = c, before solving. (ii) 6x = 52 − 2y 70 − 7y = 5x Let x = 7: 6x + 2y = 52 6(7) + 2y = 52 42 + 2y = 52 (Subtract 42 from both sides) 2y = 10 (Divide both sides by 5) y = = 5

  22. 2. Use a suitable method to solve the following simultaneous equations.In some cases it may be necessary to write the equation in the form ax + by = c, before solving. (iii) 6x = 5y 4x = 5y – 10 6x = 5y 6x – 5y = 0 4x = 5y – 10 4x– 5y = – 10 (Multiply both sides by − 1) 6x – 5y = 0 − 4x + 5y = 10 (Add rows) 2x = 10 (Divide both sides by 2) x = 5

  23. 2. Use a suitable method to solve the following simultaneous equations.In some cases it may be necessary to write the equation in the form ax + by = c, before solving. (iii) 6x = 5y 4x = 5y – 10 Let x = 5: 6x – 5y = 0 6(5) – 5y = 0 30 – 5y = 0 30 = 5y = y 6 = y

  24. 2. Use a suitable method to solve the following simultaneous equations.In some cases it may be necessary to write the equation in the form ax + by = c, before solving. (iv) 4x − 5 = 3y 2y + 1 = −2x 4x – 5 = 3y 4x – 3y = 5 (Multiply both sides by 2) 2y + 1 = –2x2x + 2y = –1 (Multiply both sides by 3) 8x – 6y = 10 6x + 6y = −3 (Add rows) 14x = 7 (Divide both sides by 14)

  25. 2. Use a suitable method to solve the following simultaneous equations.In some cases it may be necessary to write the equation in the form ax + by = c, before solving. (iv) 4x − 5 = 3y 2y + 1 = −2x Let : 4x – 3y = 5 4 – 3y = 5 2 – 3y = 5 (Add 3y to both sides) 2 = 5 + 3y (Subtract 5 from both sides) –3 = 3y –1 = y

  26. 2. Use a suitable method to solve the following simultaneous equations.In some cases it may be necessary to write the equation in the form ax + by = c, before solving. (v) 4y = 5 − 5x 3x + 34 = 5y 4y = 5 – 5x → 5x + 4y = 5 (Multiply both sides by 5) 3x + 34 = 5y → 3x − 5y = −34 (Multiply both sides by 4) 25x + 20y = 25 12x − 20y = −136 37x = –111 (Divide both sides by 37) x = –3

  27. 2. Use a suitable method to solve the following simultaneous equations.In some cases it may be necessary to write the equation in the form ax + by = c, before solving. (v) 4y = 5 − 5x 3x + 34 = 5y Let x = –3: 4y = 5 – 5x 4y = 5 – 5(–3) 4y = 5 + 15 4y = 20 (Divide both sides by 4)

  28. 2. Use a suitable method to solve the following simultaneous equations.In some cases it may be necessary to write the equation in the form ax + by = c, before solving. (vi) 2y = 7x – 13 4x = 13 + 3y 2y = 7x – 13  7x – 2y = 13 (Multiply both sides by 3) 4x = 13 + 3y4x – 3y = 13 (Multiply both sides by − 2) 21x – 6y = 39 – 8x + 6y = –26 13x = 13 (Divide both sides by 13) x = 1

  29. 2. Use a suitable method to solve the following simultaneous equations.In some cases it may be necessary to write the equation in the form ax + by = c, before solving. (vi) 2y = 7x – 13 4x = 13 + 3y Let x = 1: 2y = 7x – 13 2y = 7(1) – 13 2y = 7 – 13 2y = – 6 (Divide both sides by 2) y = y = –3

  30. 3. Solve the following simultaneous equations, involving fractions. (i) 2x − y = 18

  31. 3. Solve the following simultaneous equations, involving fractions. (i) 2x − y = 18 Let x = 15: 2x − y = 18 2(15) – y = 18 30 – y = 18 (Add y to both sides) 30 = 18 + y (Subtract 18 from both sides) 12 = y

  32. 3. Solve the following simultaneous equations, involving fractions. (ii)

  33. 3. Solve the following simultaneous equations, involving fractions. (ii) Let a = 2:

  34. 3. Solve the following simultaneous equations, involving fractions. (iii)

  35. 3. Solve the following simultaneous equations, involving fractions. (iii) Let x = 6:

  36. 3. Solve the following simultaneous equations, involving fractions. (iv)

  37. 3. Solve the following simultaneous equations, involving fractions. (iv) Let x = 10:

  38. 3. Solve the following simultaneous equations, involving fractions. (v) a + b = 2 (Multiply both sides by 3) a – 3b = 6 3a + 3b = 6 a – 3b = 6 4a = 12 (Divide both sides by 4) a = 3

  39. 3. Solve the following simultaneous equations, involving fractions. (v) Let a = 3: 3 + b = 2 (Subtract 3 from both sides) b = – 1

  40. 3. Solve the following simultaneous equations, involving fractions. (vi) 5(2p – 5) + 3(q) = 90 3p + 2(10) = 5(3q – 5) 10p – 25 + 3q = 90 (Add 25 to both sides) 3p + 20 = 15q – 25 (Subtract 20 and 15q from both sides)

  41. 3. Solve the following simultaneous equations, involving fractions. (vi) 10p + 3q = 115 (Multiply both sides by 5) 3p – 15q = – 45 50p + 15q = 575 3p – 15q = – 45 53p = 530 (Divide both sides by 53) p = 10

  42. 3. Solve the following simultaneous equations, involving fractions. (vi) Let p = 10:

  43. 4. The diagram shows a rectangle. (i) Form two equations, in terms of x and y. Opposite sides of a rectangle are equal. So, 5x – 2y = 9 and x – 2y = 5

  44. 4. The diagram shows a rectangle. (ii) Solve these equations to find the values of x and y. 5x – 2y = 9 x – 2y = 5 (Multiply both sides by − 1) 5x – 2y = 9 –x + 2y = –5 (Add Rows) 4x = 4 (Divide both sides by 4) x = 1

  45. 4. The diagram shows a rectangle. (ii) Solve these equations to find the values of x and y. Let x = 1: x – 2y = 5 1 – 2y = 5 (Subtract 1 from both sides) – 2y = 4 (Divide both sides by − 2) y = – 2

  46. 4. The diagram shows a rectangle. (iii) Hence, verify that the perimeter of the rectangle is 28 units. Perimeter = (x – 2y) + (5x – 2y) + 5 + 9 Perimeter = (1 – 2(–2)) + (5(1) – 2(–2)) + 5 + 9 Perimeter = (1 + 4) + (5 + 4) + 5 + 9 Perimeter = 5 + 9 + 5 + 9 Perimeter = 28 units

  47. 5. The diagram shows a parallelogram. (i) Form two equations, in terms of x and y. Opposite sides of a parallelogram are equal: 2x – 5y = 11 and 3x + 2y = 7

  48. 5. The diagram shows a parallelogram. (ii) Solve these equations to find the values of x and y. 2x – 5y = 11 (Multiply both sides by 2) 3x + 2y = 7 (Multiply both sides by 5) 4x – 10y = 22 15x + 10y = 35 (Add rows) 19x = 57 (Divide both sides by 19) x = 3

  49. 5. The diagram shows a parallelogram. (ii) Solve these equations to find the values of x and y. Let x = 3: 2x – 5y = 11 2(3) – 5y = 11 6 – 5y = 11 (Subtract 6 from both sides) – 5y = 5 (Divide both sides by – 5) y = – 1

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