Mathematics 10 — Exercise 1.4 · Q2 (ii)

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.

   $$(w\times \left(-i-3\right)+2\times z=9-i,-i\times w+i\times z=i-1)$$

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

   $$(-i\times \left(w\times \left(-i-3\right)+2\times z\right)=-i\times \left(9-i\right),\left(-i-3\right)\times \left(-i\times w+i\times z\right)=\left(-i-3\right)\times \left(i-1\right))$$

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

   $$z\times \left(i-1\right)=-7\times i-5$$

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

   $$\frac{-7\times i-5}{i-1}$$

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

   $$-i-1$$

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

   $$\left(\frac{-7\times i-5}{i-1}\right)\times \left(\frac{-i-1}{-i-1}\right)$$

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

   $$\frac{\left(-7\times i-5\right)\times \left(-i-1\right)}{\left(-i-1\right)\times \left(i-1\right)}$$

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.

   $$-7\times i\times \left(-i-1\right)-5\times \left(-i-1\right)$$

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

    $$--5-(-5\times i)-7\times \left(-1\right)\times i-7\times \left(-1\right)\times ii$$

11. Real × real: multiply the numerical coefficients.

    $$\left(-5\right)\times \left(-1\right)=5$$

12. Real × imaginary: multiply the numerical coefficients.

    $$--5\times i=5\times i$$

13. Imaginary × real: multiply the numerical coefficients.

    $$-7\times \left(-1\right)\times i=7\times i$$

14. Imaginary × imaginary: multiply the numerical coefficients.

    $$-7\times \left(-1\right)\times ii=7\times i^2$$

15. Substitute these four products into the expansion.

    $$5+5\times i+7\times i+7\times i^2$$

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

    $$7\times i^2=7\times \left(-1\right)$$

17. Multiply its coefficient by −1.

    $$7\times \left(-1\right)=-7$$

18. Rewrite the expansion after this sign calculation.

    $$5+5\times i+7\times i-7$$

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

    $$5-7+i\times \left(5+7\right)$$

20. Calculate the sum of the real terms.

    $$5-7=-2$$

21. Calculate the sum of the imaginary coefficients.

    $$5+7=12$$

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

    $$12\times i-2$$

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.

    $$-(-i-1)+i\times \left(-i-1\right)$$

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

    $$--1-(-i)+\left(-1\right)\times i+\left(-1\right)\times ii$$

26. Real × real: multiply the numerical coefficients.

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

27. Real × imaginary: multiply the numerical coefficients.

    $$--1\times i=i$$

28. Imaginary × real: multiply the numerical coefficients.

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

29. Imaginary × imaginary: multiply the numerical coefficients.

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

30. Substitute these four products into the expansion.

    $$1+i-i-i^2$$

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

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

32. Multiply its coefficient by −1.

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

33. Rewrite the expansion after this sign calculation.

    $$1+i-i+1$$

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

    $$1+1+i\times \left(1-1\right)$$

35. Calculate the sum of the real terms.

    $$1+1=2$$

36. Calculate the sum of the imaginary coefficients.

    $$1-1=0$$

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

    $$0\times i+2$$

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

    $$2=2$$

39. Put the calculated numerator above the calculated denominator.

    $$\frac{12\times i-2}{2}$$

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

    $$-\frac{2}{2}+\frac{12\times i}{2}$$

41. Move i outside its numerical fraction.

    $$-\frac{2}{2}+\left(\frac{12}{2}\right)\times i$$

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

    $$\frac{\left(-2\right)\times 1}{1\times 2}$$

43. Calculate these two products separately.

    $$-\frac{2}{2}$$

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

    $$\mathrm{GCD}=2$$

45. Show the two divisions explicitly.

    $$\frac{-\frac{2}{2}}{\frac{2}{2}}$$

46. Evaluate the denominator division.

    $$\frac{2}{2}=1$$

47. Write the reduced fraction.

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

48. Calculate this operation before combining the remaining terms.

    $$\left(-2\right)\times \frac{1}{2}=-1$$

49. Reduce the real coefficient by dividing numerator and denominator by 2.

    $$-\frac{2}{2}=\frac{-\frac{2}{2}}{\frac{2}{2}}$$

50. Simplify the real coefficient fraction.

    $$-\frac{2}{2}=-1$$

51. Reduce the imaginary coefficient by dividing numerator and denominator by 2.

    $$\frac{12}{2}=\frac{\frac{12}{2}}{\frac{2}{2}}$$

52. Simplify the imaginary coefficient fraction.

    $$\frac{12}{2}=6$$

53. Combine the reduced real and imaginary terms.

    $$6\times i-1$$

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

    $$12\times i+w\times \left(-i-3\right)-2=9-i$$

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

    $$w\times \left(-i-3\right)=11-13\times i$$

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

    $$\frac{11-13\times i}{-i-3}$$

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

    $$i-3$$

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

    $$\left(\frac{11-13\times i}{-i-3}\right)\times \left(\frac{i-3}{i-3}\right)$$

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

    $$\frac{\left(11-13\times i\right)\times \left(i-3\right)}{\left(-i-3\right)\times \left(i-3\right)}$$

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.

    $$-13\times i\times \left(i-3\right)+11\times \left(i-3\right)$$

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

    $$11\times \left(-3\right)+11\times 1\times i-13\times \left(-3\right)\times i-13\times 1\times ii$$

63. Real × real: multiply the numerical coefficients.

    $$11\times \left(-3\right)=-33$$

64. Real × imaginary: multiply the numerical coefficients.

    $$11\times 1\times i=11\times i$$

65. Imaginary × real: multiply the numerical coefficients.

    $$-13\times \left(-3\right)\times i=39\times i$$

66. Imaginary × imaginary: multiply the numerical coefficients.

    $$-13\times 1\times ii=-13\times i^2$$

67. Substitute these four products into the expansion.

    $$-33+11\times i+39\times i-13\times i^2$$

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

    $$-13\times i^2=\left(-13\right)\times \left(-1\right)$$

69. Multiply its coefficient by −1.

    $$\left(-13\right)\times \left(-1\right)=13$$

70. Rewrite the expansion after this sign calculation.

    $$-33+11\times i+39\times i+13$$

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

    $$-33+13+i\times \left(11+39\right)$$

72. Calculate the sum of the real terms.

    $$-33+13=-20$$

73. Calculate the sum of the imaginary coefficients.

    $$11+39=50$$

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

    $$50\times i-20$$

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.

    $$-i\times \left(i-3\right)-3\times \left(i-3\right)$$

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

    $$--9-3\times i-\left(-3\right)\times i-1\times ii$$

78. Real × real: multiply the numerical coefficients.

    $$\left(-3\right)\times \left(-3\right)=9$$

79. Real × imaginary: multiply the numerical coefficients.

    $$-3\times i=-3\times i$$

80. Imaginary × real: multiply the numerical coefficients.

    $$-\left(-3\right)\times i=3\times i$$

81. Imaginary × imaginary: multiply the numerical coefficients.

    $$-1\times ii=-i^2$$

82. Substitute these four products into the expansion.

    $$9-3\times i+3\times i-i^2$$

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

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

84. Multiply its coefficient by −1.

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

85. Rewrite the expansion after this sign calculation.

    $$9-3\times i+3\times i+1$$

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

    $$9+1+i\times \left(-3+3\right)$$

87. Calculate the sum of the real terms.

    $$9+1=10$$

88. Calculate the sum of the imaginary coefficients.

    $$-3+3=0$$

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

    $$0\times i+10$$

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

    $$10=10$$

91. Put the calculated numerator above the calculated denominator.

    $$\frac{50\times i-20}{10}$$

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

    $$-\frac{20}{10}+\frac{50\times i}{10}$$

93. Move i outside its numerical fraction.

    $$-\frac{20}{10}+\left(\frac{50}{10}\right)\times i$$

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

    $$\frac{\left(-20\right)\times 1}{1\times 10}$$

95. Calculate these two products separately.

    $$-\frac{20}{10}$$

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

    $$\mathrm{GCD}=10$$

97. Show the two divisions explicitly.

    $$\frac{-\frac{20}{10}}{\frac{10}{10}}$$

98. Evaluate the denominator division.

    $$\frac{10}{10}=1$$

99. Write the reduced fraction.

    $$-2\times \frac{1}{1}$$

100. Calculate this operation before combining the remaining terms.

     $$\left(-20\right)\times \frac{1}{10}=-2$$

101. Reduce the real coefficient by dividing numerator and denominator by 10.

     $$-\frac{20}{10}=\frac{-\frac{20}{10}}{\frac{10}{10}}$$

102. Simplify the real coefficient fraction.

     $$-\frac{20}{10}=-2$$

103. Reduce the imaginary coefficient by dividing numerator and denominator by 10.

     $$\frac{50}{10}=\frac{\frac{50}{10}}{\frac{10}{10}}$$

104. Simplify the imaginary coefficient fraction.

     $$\frac{50}{10}=5$$

105. Combine the reduced real and imaginary terms.

     $$5\times i-2$$

106. Check z and w in both original equations.

     $$(9-i=9-i,i-1=i-1)$$

Answer

The final answer is shown below.