Mathematics 10 — Exercise 1.4 · Q2 (iii)

Full question

Solve the simultaneous equations for complex numbers z and w.

Step-by-step solution

1. We have two equations for the two unknown complex numbers z and w.

   $$(-4\times w+z=3\times i,3\times w+2\times z=11-5\times i)$$

2. Multiply the first equation by the coefficient of w in the second, and the second by the coefficient of w in the first.

   $$(-12\times w+3\times z=9\times i,-12\times w-8\times z=20\times i-44)$$

3. Subtract. Both w terms cancel. Expand the remaining coefficients using i² = −1.

   $$11\times z=44-11\times i$$

4. Start with the original quotient. For an inverse, the numerator is 1.

   $$\frac{44-11\times i}{11}$$

5. Find the conjugate of the denominator: keep its real part and reverse the imaginary sign.

   $$11$$

6. Multiply by the conjugate over itself; this fraction equals 1.

   $$\left(\frac{11}{11}\right)\times \left(\frac{44-11\times i}{11}\right)$$

7. Multiply numerators together and denominators together. Keep their brackets intact.

   $$\frac{11\times \left(44-11\times i\right)}{11\times 11}$$

8. Calculate the numerator first.

9. Distribute the first bracket: multiply its real term by the whole second bracket, then its imaginary term by that bracket.

   $$44\times 11-11\times 11\times i$$

10. Distribute each of those products. Keep all four multiplications visible.

    $$44\times 11+44\times 0\times i-11\times 11\times i-11\times 0\times ii$$

11. Real × real: multiply the numerical coefficients.

    $$44\times 11=484$$

12. Real × imaginary: multiply the numerical coefficients.

    $$44\times 0\times i=0\times i$$

13. Imaginary × real: multiply the numerical coefficients.

    $$-11\times 11\times i=-121\times i$$

14. Imaginary × imaginary: multiply the numerical coefficients.

    $$-11\times 0\times ii=0\times i^2$$

15. Substitute these four products into the expansion.

    $$484+0\times i-121\times i+0\times i^2$$

16. In the last term replace i² by −1.

    $$0\times i^2=0\times \left(-1\right)$$

17. Multiply its coefficient by −1.

    $$0\times \left(-1\right)=0$$

18. Rewrite the expansion after this sign calculation.

    $$484+0\times i-121\times i+0$$

19. Group the real terms and factor i out of the imaginary terms.

    $$484+0+i\times \left(0-121\right)$$

20. Calculate the sum of the real terms.

    $$484+0=484$$

21. Calculate the sum of the imaginary coefficients.

    $$0-121=-121$$

22. Insert both calculated sums in a + bi form.

    $$484-121\times i$$

23. Now calculate the denominator independently.

24. Distribute the first bracket: multiply its real term by the whole second bracket, then its imaginary term by that bracket.

    $$11\times 11+0\times 11\times i$$

25. Distribute each of those products. Keep all four multiplications visible.

    $$11\times 11+11\times 0\times i+0\times 11\times i+0\times 0\times ii$$

26. Real × real: multiply the numerical coefficients.

    $$11\times 11=121$$

27. Real × imaginary: multiply the numerical coefficients.

    $$11\times 0\times i=0\times i$$

28. Imaginary × real: multiply the numerical coefficients.

    $$0\times 11\times i=0\times i$$

29. Imaginary × imaginary: multiply the numerical coefficients.

    $$0\times 0\times ii=0\times i^2$$

30. Substitute these four products into the expansion.

    $$121+0\times i+0\times i+0\times i^2$$

31. In the last term replace i² by −1.

    $$0\times i^2=0\times \left(-1\right)$$

32. Multiply its coefficient by −1.

    $$0\times \left(-1\right)=0$$

33. Rewrite the expansion after this sign calculation.

    $$121+0\times i+0\times i+0$$

34. Group the real terms and factor i out of the imaginary terms.

    $$121+0+i\times \left(0+0\right)$$

35. Calculate the sum of the real terms.

    $$121+0=121$$

36. Calculate the sum of the imaginary coefficients.

    $$0+0=0$$

37. Insert both calculated sums in a + bi form.

    $$0\times i+121$$

38. The imaginary coefficient in the denominator is zero, so the denominator is real.

    $$121=121$$

39. Put the calculated numerator above the calculated denominator.

    $$\frac{484-121\times i}{121}$$

40. Separate the real term and imaginary term into two fractions with the same denominator.

    $$\frac{484}{121}+\frac{-121\times i}{121}$$

41. Move i outside its numerical fraction.

    $$\frac{484}{121}+\left(-\frac{121}{121}\right)\times i$$

42. Reduce the real coefficient by dividing numerator and denominator by 121.

    $$\frac{484}{121}=\frac{\frac{484}{121}}{\frac{121}{121}}$$

43. Simplify the real coefficient fraction.

    $$\frac{484}{121}=4$$

44. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-121\right)\times 1}{1\times 121}$$

45. Calculate these two products separately.

    $$-\frac{121}{121}$$

46. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=121$$

47. Show the two divisions explicitly.

    $$\frac{-\frac{121}{121}}{\frac{121}{121}}$$

48. Evaluate the denominator division.

    $$\frac{121}{121}=1$$

49. Write the reduced fraction.

    $$-\frac{1}{1}$$

50. Calculate this operation before combining the remaining terms.

    $$\left(-121\right)\times \frac{1}{121}=-1$$

51. Reduce the imaginary coefficient by dividing numerator and denominator by 121.

    $$-\frac{121}{121}=\frac{-\frac{121}{121}}{\frac{121}{121}}$$

52. Simplify the imaginary coefficient fraction.

    $$-\frac{121}{121}=-1$$

53. Combine the reduced real and imaginary terms.

    $$4-i$$

54. This quotient is z. Substitute it into the first original equation.

    $$-i-4\times w+4=3\times i$$

55. Subtract the known z term from both sides, then divide by the coefficient of w.

    $$-4\times w=4\times i-4$$

56. Start with the original quotient. For an inverse, the numerator is 1.

    $$\frac{4\times i-4}{-4}$$

57. Find the conjugate of the denominator: keep its real part and reverse the imaginary sign.

    $$-4$$

58. Multiply by the conjugate over itself; this fraction equals 1.

    $$\left(-\frac{4}{-4}\right)\times \left(\frac{4\times i-4}{-4}\right)$$

59. Multiply numerators together and denominators together. Keep their brackets intact.

    $$\frac{\left(-4\right)\times \left(4\times i-4\right)}{\left(-4\right)\times \left(-4\right)}$$

60. Calculate the numerator first.

61. Distribute the first bracket: multiply its real term by the whole second bracket, then its imaginary term by that bracket.

    $$--16+4\times \left(-4\right)\times i$$

62. Distribute each of those products. Keep all four multiplications visible.

    $$--16-0\times i+4\times \left(-4\right)\times i+4\times 0\times ii$$

63. Real × real: multiply the numerical coefficients.

    $$\left(-4\right)\times \left(-4\right)=16$$

64. Real × imaginary: multiply the numerical coefficients.

    $$-0\times i=0\times i$$

65. Imaginary × real: multiply the numerical coefficients.

    $$4\times \left(-4\right)\times i=-16\times i$$

66. Imaginary × imaginary: multiply the numerical coefficients.

    $$4\times 0\times ii=0\times i^2$$

67. Substitute these four products into the expansion.

    $$16+0\times i-16\times i+0\times i^2$$

68. In the last term replace i² by −1.

    $$0\times i^2=0\times \left(-1\right)$$

69. Multiply its coefficient by −1.

    $$0\times \left(-1\right)=0$$

70. Rewrite the expansion after this sign calculation.

    $$16+0\times i-16\times i+0$$

71. Group the real terms and factor i out of the imaginary terms.

    $$16+0+i\times \left(0-16\right)$$

72. Calculate the sum of the real terms.

    $$16+0=16$$

73. Calculate the sum of the imaginary coefficients.

    $$0-16=-16$$

74. Insert both calculated sums in a + bi form.

    $$16-16\times i$$

75. Now calculate the denominator independently.

76. Distribute the first bracket: multiply its real term by the whole second bracket, then its imaginary term by that bracket.

    $$--16+0\times \left(-4\right)\times i$$

77. Distribute each of those products. Keep all four multiplications visible.

    $$--16-0\times i+0\times \left(-4\right)\times i+0\times 0\times ii$$

78. Real × real: multiply the numerical coefficients.

    $$\left(-4\right)\times \left(-4\right)=16$$

79. Real × imaginary: multiply the numerical coefficients.

    $$-0\times i=0\times i$$

80. Imaginary × real: multiply the numerical coefficients.

    $$0\times \left(-4\right)\times i=0\times i$$

81. Imaginary × imaginary: multiply the numerical coefficients.

    $$0\times 0\times ii=0\times i^2$$

82. Substitute these four products into the expansion.

    $$16+0\times i+0\times i+0\times i^2$$

83. In the last term replace i² by −1.

    $$0\times i^2=0\times \left(-1\right)$$

84. Multiply its coefficient by −1.

    $$0\times \left(-1\right)=0$$

85. Rewrite the expansion after this sign calculation.

    $$16+0\times i+0\times i+0$$

86. Group the real terms and factor i out of the imaginary terms.

    $$16+0+i\times \left(0+0\right)$$

87. Calculate the sum of the real terms.

    $$16+0=16$$

88. Calculate the sum of the imaginary coefficients.

    $$0+0=0$$

89. Insert both calculated sums in a + bi form.

    $$0\times i+16$$

90. The imaginary coefficient in the denominator is zero, so the denominator is real.

    $$16=16$$

91. Put the calculated numerator above the calculated denominator.

    $$\frac{16-16\times i}{16}$$

92. Separate the real term and imaginary term into two fractions with the same denominator.

    $$\frac{16}{16}+\frac{-16\times i}{16}$$

93. Move i outside its numerical fraction.

    $$\frac{16}{16}+\left(-\frac{16}{16}\right)\times i$$

94. Reduce the real coefficient by dividing numerator and denominator by 16.

    $$\frac{16}{16}=\frac{\frac{16}{16}}{\frac{16}{16}}$$

95. Simplify the real coefficient fraction.

    $$\frac{16}{16}=1$$

96. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-16\right)\times 1}{1\times 16}$$

97. Calculate these two products separately.

    $$-\frac{16}{16}$$

98. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=16$$

99. Show the two divisions explicitly.

    $$\frac{-\frac{16}{16}}{\frac{16}{16}}$$

100. Calculate this operation before combining the remaining terms.

     $$\left(-16\right)\times \frac{1}{16}=-1$$

101. Reduce the imaginary coefficient by dividing numerator and denominator by 16.

     $$-\frac{16}{16}=\frac{-\frac{16}{16}}{\frac{16}{16}}$$

102. Simplify the imaginary coefficient fraction.

     $$-\frac{16}{16}=-1$$

103. Combine the reduced real and imaginary terms.

     $$1-i$$

104. Check z and w in both original equations.

     $$(3\times i=3\times i,11-5\times i=11-5\times i)$$

Answer

The final answer is shown below.