Mathematics 10 — Exercise 1.4 · Q2 (v)

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.

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

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

   $$(\left(i-1\right)\times \left(w\times \left(i+3\right)+2\times z\right)=i-1,\left(i+3\right)\times \left(w\times \left(i-1\right)-z\right)=2\times i+6)$$

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

   $$z\times \left(3\times i+1\right)=-i-7$$

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

   $$\frac{-i-7}{3\times i+1}$$

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

   $$1-3\times i$$

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

   $$\left(\frac{1-3\times i}{1-3\times i}\right)\times \left(\frac{-i-7}{3\times i+1}\right)$$

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

   $$\frac{\left(1-3\times i\right)\times \left(-i-7\right)}{\left(1-3\times i\right)\times \left(3\times i+1\right)}$$

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.

   $$-i\times \left(1-3\times i\right)-7\times \left(1-3\times i\right)$$

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

    $$-7-(-21\times i)-1\times i-\left(-3\right)\times ii$$

11. Real × real: multiply the numerical coefficients.

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

12. Real × imaginary: multiply the numerical coefficients.

    $$--21\times i=21\times i$$

13. Imaginary × real: multiply the numerical coefficients.

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

14. Imaginary × imaginary: multiply the numerical coefficients.

    $$-\left(-3\right)\times ii=3\times i^2$$

15. Substitute these four products into the expansion.

    $$-7+21\times i-i+3\times i^2$$

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

    $$3\times i^2=3\times \left(-1\right)$$

17. Multiply its coefficient by −1.

    $$3\times \left(-1\right)=-3$$

18. Rewrite the expansion after this sign calculation.

    $$-7+21\times i-i-3$$

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

    $$-10+i\times \left(21-1\right)$$

20. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 3=-3$$

21. Calculate the sum of the real terms.

    $$-7-3=-10$$

22. Calculate the sum of the imaginary coefficients.

    $$21-1=20$$

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

    $$20\times i-10$$

24. Now calculate the denominator independently.

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

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

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

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

27. Real × real: multiply the numerical coefficients.

    $$1\times 1=1$$

28. Real × imaginary: multiply the numerical coefficients.

    $$\left(-1\right)\times 3\times i=-3\times i$$

29. Imaginary × real: multiply the numerical coefficients.

    $$3\times 1\times i=3\times i$$

30. Imaginary × imaginary: multiply the numerical coefficients.

    $$3\times \left(-3\right)\times ii=-9\times i^2$$

31. Substitute these four products into the expansion.

    $$1-3\times i+3\times i-9\times i^2$$

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

    $$-9\times i^2=\left(-9\right)\times \left(-1\right)$$

33. Multiply its coefficient by −1.

    $$\left(-9\right)\times \left(-1\right)=9$$

34. Rewrite the expansion after this sign calculation.

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

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

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

36. Calculate the sum of the real terms.

    $$1+9=10$$

37. Calculate the sum of the imaginary coefficients.

    $$-3+3=0$$

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

    $$0\times i+10$$

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

    $$10=10$$

40. Put the calculated numerator above the calculated denominator.

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

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

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

42. Move i outside its numerical fraction.

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

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

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

44. Calculate these two products separately.

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

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

    $$\mathrm{GCD}=10$$

46. Show the two divisions explicitly.

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

47. Evaluate the denominator division.

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

48. Write the reduced fraction.

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

49. Calculate this operation before combining the remaining terms.

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

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

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

51. Simplify the real coefficient fraction.

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

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

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

53. Simplify the imaginary coefficient fraction.

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

54. Combine the reduced real and imaginary terms.

    $$2\times i-1$$

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

    $$4\times i+w\times \left(i+3\right)-2=1$$

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

    $$w\times \left(i+3\right)=3-4\times i$$

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

    $$\frac{3-4\times i}{i+3}$$

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

    $$3-i$$

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

    $$\left(\frac{3-4\times i}{i+3}\right)\times \left(\frac{3-i}{3-i}\right)$$

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

    $$\frac{\left(3-4\times i\right)\times \left(3-i\right)}{\left(3-i\right)\times \left(i+3\right)}$$

61. Calculate the numerator first.

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

    $$-4\times i\times \left(3-i\right)+3\times \left(3-i\right)$$

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

    $$3\times 3+3\times \left(-1\right)\times i-4\times 3\times i-4\times \left(-1\right)\times ii$$

64. Real × real: multiply the numerical coefficients.

    $$3\times 3=9$$

65. Real × imaginary: multiply the numerical coefficients.

    $$3\times \left(-1\right)\times i=-3\times i$$

66. Imaginary × real: multiply the numerical coefficients.

    $$-4\times 3\times i=-12\times i$$

67. Imaginary × imaginary: multiply the numerical coefficients.

    $$-4\times \left(-1\right)\times ii=4\times i^2$$

68. Substitute these four products into the expansion.

    $$9-3\times i-12\times i+4\times i^2$$

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

    $$4\times i^2=4\times \left(-1\right)$$

70. Multiply its coefficient by −1.

    $$4\times \left(-1\right)=-4$$

71. Rewrite the expansion after this sign calculation.

    $$9-3\times i-12\times i-4$$

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

    $$9-4+i\times \left(-3-12\right)$$

73. Calculate the sum of the real terms.

    $$9-4=5$$

74. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 12=-12$$

75. Calculate the sum of the imaginary coefficients.

    $$-3-12=-15$$

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

    $$5-15\times i$$

77. Now calculate the denominator independently.

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

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

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

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

80. Real × real: multiply the numerical coefficients.

    $$3\times 3=9$$

81. Real × imaginary: multiply the numerical coefficients.

    $$3\times \left(-1\right)\times i=-3\times i$$

82. Imaginary × real: multiply the numerical coefficients.

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

83. Imaginary × imaginary: multiply the numerical coefficients.

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

84. Substitute these four products into the expansion.

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

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

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

86. Multiply its coefficient by −1.

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

87. Rewrite the expansion after this sign calculation.

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

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

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

89. Calculate the sum of the real terms.

    $$9+1=10$$

90. Calculate the sum of the imaginary coefficients.

    $$-3+3=0$$

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

    $$0\times i+10$$

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

    $$10=10$$

93. Put the calculated numerator above the calculated denominator.

    $$\frac{5-15\times i}{10}$$

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

    $$\frac{5}{10}+\frac{-15\times i}{10}$$

95. Move i outside its numerical fraction.

    $$\frac{5}{10}+\left(-\frac{15}{10}\right)\times i$$

96. Reduce the real coefficient by dividing numerator and denominator by 5.

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

97. Simplify the real coefficient fraction.

    $$\frac{5}{10}=\frac{1}{2}$$

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

    $$\frac{\left(-15\right)\times 1}{1\times 10}$$

99. Calculate these two products separately.

    $$-\frac{15}{10}$$

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

     $$\mathrm{GCD}=5$$

101. Show the two divisions explicitly.

     $$\frac{-\frac{15}{5}}{\frac{10}{5}}$$

102. Evaluate the numerator division.

     $$-\frac{15}{5}=-3$$

103. Evaluate the denominator division.

     $$\frac{10}{5}=2$$

104. Write the reduced fraction.

     $$-\frac{3}{2}$$

105. Calculate this operation before combining the remaining terms.

     $$\left(-15\right)\times \frac{1}{10}=-\frac{3}{2}$$

106. Reduce the imaginary coefficient by dividing numerator and denominator by 5.

     $$-\frac{15}{10}=\frac{-\frac{15}{5}}{\frac{10}{5}}$$

107. Simplify the imaginary coefficient fraction.

     $$-\frac{15}{10}=-\frac{3}{2}$$

108. Combine the reduced real and imaginary terms.

     $$\frac{1}{2}-\frac{3\times i}{2}$$

109. Check z and w in both original equations.

     $$(1=1,2=2)$$

Answer

The final answer is shown below.