Mathematics 10 — Exercise 1.4 · Q2 (i)

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.

   $$(w\times \left(i+2\right)+3\times z=11-i,-w+z\times \left(2-i\right)=i-1)$$

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

   $$(-w\times \left(i+2\right)-3\times z=i-11,\left(i+2\right)\times \left(-w+z\times \left(2-i\right)\right)=\left(i-1\right)\times \left(i+2\right))$$

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

   $$-8\times z=-8$$

4. Calculate this operation before combining the remaining terms.

   $$\frac{1}{-8}=-\frac{1}{8}$$

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

   $$\frac{\left(-8\right)\times \left(-1\right)}{1\times 8}$$

6. Calculate these two products separately.

   $$\frac{8}{8}$$

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

   $$\mathrm{GCD}=8$$

8. Show the two divisions explicitly.

   $$\frac{\frac{8}{8}}{\frac{8}{8}}$$

9. Evaluate the numerator division.

   $$\frac{8}{8}=1$$

10. Write the reduced fraction.

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

11. Calculate this operation before combining the remaining terms.

    $$\left(-8\right)\times \left(-\frac{1}{8}\right)=1$$

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

    $$-\frac{8}{-8}$$

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

    $$-8$$

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

    $$-\frac{8}{-8}\times -\frac{8}{-8}$$

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

    $$\frac{64\times 1}{1\times 64}$$

16. Calculate these two products separately.

    $$\frac{64}{64}$$

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

    $$\mathrm{GCD}=64$$

18. Show the two divisions explicitly.

    $$\frac{\frac{64}{64}}{\frac{64}{64}}$$

19. Calculate this operation before combining the remaining terms.

    $$64\times \frac{1}{64}=1$$

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

    $$\frac{\left(-8\right)\times \left(-8\right)}{\left(-8\right)\times \left(-8\right)}$$

21. Calculate the numerator first.

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

    $$--64+0\times \left(-8\right)\times i$$

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

    $$--64-0\times i+0\times \left(-8\right)\times i+0\times 0\times ii$$

24. Real × real: multiply the numerical coefficients.

    $$\left(-8\right)\times \left(-8\right)=64$$

25. Real × imaginary: multiply the numerical coefficients.

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

26. Imaginary × real: multiply the numerical coefficients.

    $$0\times \left(-8\right)\times i=0\times i$$

27. Imaginary × imaginary: multiply the numerical coefficients.

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

28. Substitute these four products into the expansion.

    $$64+0\times i+0\times i+0\times i^2$$

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

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

30. Multiply its coefficient by −1.

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

31. Rewrite the expansion after this sign calculation.

    $$64+0\times i+0\times i+0$$

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

    $$64+0+i\times \left(0+0\right)$$

33. Calculate the sum of the real terms.

    $$64+0=64$$

34. Calculate the sum of the imaginary coefficients.

    $$0+0=0$$

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

    $$0\times i+64$$

36. Now calculate the denominator independently.

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

    $$--64+0\times \left(-8\right)\times i$$

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

    $$--64-0\times i+0\times \left(-8\right)\times i+0\times 0\times ii$$

39. Real × real: multiply the numerical coefficients.

    $$\left(-8\right)\times \left(-8\right)=64$$

40. Real × imaginary: multiply the numerical coefficients.

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

41. Imaginary × real: multiply the numerical coefficients.

    $$0\times \left(-8\right)\times i=0\times i$$

42. Imaginary × imaginary: multiply the numerical coefficients.

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

43. Substitute these four products into the expansion.

    $$64+0\times i+0\times i+0\times i^2$$

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

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

45. Multiply its coefficient by −1.

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

46. Rewrite the expansion after this sign calculation.

    $$64+0\times i+0\times i+0$$

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

    $$64+0+i\times \left(0+0\right)$$

48. Calculate the sum of the real terms.

    $$64+0=64$$

49. Calculate the sum of the imaginary coefficients.

    $$0+0=0$$

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

    $$0\times i+64$$

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

    $$64=64$$

52. Put the calculated numerator above the calculated denominator.

    $$\frac{0\times i+64}{64}$$

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

    $$\frac{64}{64}+\frac{0\times i}{64}$$

54. Move i outside its numerical fraction.

    $$\frac{64}{64}+\left(\frac{0}{64}\right)\times i$$

55. Reduce the real coefficient by dividing numerator and denominator by 64.

    $$\frac{64}{64}=\frac{\frac{64}{64}}{\frac{64}{64}}$$

56. Simplify the real coefficient fraction.

    $$\frac{64}{64}=1$$

57. Reduce the imaginary coefficient by dividing numerator and denominator by 64.

    $$\frac{0}{64}=\frac{\frac{0}{64}}{\frac{64}{64}}$$

58. Simplify the imaginary coefficient fraction.

    $$\frac{0}{64}=0$$

59. Combine the reduced real and imaginary terms.

    $$0\times i+1$$

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

    $$w\times \left(i+2\right)+3=11-i$$

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

    $$w\times \left(i+2\right)=8-i$$

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

    $$\frac{8-i}{i+2}$$

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

    $$2-i$$

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

    $$\left(\frac{2-i}{2-i}\right)\times \left(\frac{8-i}{i+2}\right)$$

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

    $$\frac{\left(2-i\right)\times \left(8-i\right)}{\left(2-i\right)\times \left(i+2\right)}$$

66. Calculate the numerator first.

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

    $$-i\times \left(2-i\right)+8\times \left(2-i\right)$$

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

    $$8\times 2+8\times \left(-1\right)\times i-2\times i-\left(-1\right)\times ii$$

69. Real × real: multiply the numerical coefficients.

    $$8\times 2=16$$

70. Real × imaginary: multiply the numerical coefficients.

    $$8\times \left(-1\right)\times i=-8\times i$$

71. Imaginary × real: multiply the numerical coefficients.

    $$-2\times i=-2\times i$$

72. Imaginary × imaginary: multiply the numerical coefficients.

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

73. Substitute these four products into the expansion.

    $$16-8\times i-2\times i+i^2$$

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

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

75. Multiply its coefficient by −1.

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

76. Rewrite the expansion after this sign calculation.

    $$16-8\times i-2\times i-1$$

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

    $$16-1+i\times \left(-8-2\right)$$

78. Calculate the sum of the real terms.

    $$16-1=15$$

79. Calculate this operation before combining the remaining terms.

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

80. Calculate the sum of the imaginary coefficients.

    $$-8-2=-10$$

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

    $$15-10\times i$$

82. Now calculate the denominator independently.

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

    $$2\times \left(2-i\right)+i\times \left(2-i\right)$$

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

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

85. Real × real: multiply the numerical coefficients.

    $$2\times 2=4$$

86. Real × imaginary: multiply the numerical coefficients.

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

87. Imaginary × real: multiply the numerical coefficients.

    $$2\times i=2\times i$$

88. Imaginary × imaginary: multiply the numerical coefficients.

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

89. Substitute these four products into the expansion.

    $$4-2\times i+2\times i-i^2$$

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

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

91. Multiply its coefficient by −1.

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

92. Rewrite the expansion after this sign calculation.

    $$4-2\times i+2\times i+1$$

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

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

94. Calculate the sum of the real terms.

    $$4+1=5$$

95. Calculate the sum of the imaginary coefficients.

    $$-2+2=0$$

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

    $$0\times i+5$$

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

    $$5=5$$

98. Put the calculated numerator above the calculated denominator.

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

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

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

100. Move i outside its numerical fraction.

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

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

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

102. Simplify the real coefficient fraction.

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

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

     $$\frac{\left(-10\right)\times 1}{1\times 5}$$

104. Calculate these two products separately.

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

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

     $$\mathrm{GCD}=5$$

106. Show the two divisions explicitly.

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

107. Evaluate the denominator division.

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

108. Write the reduced fraction.

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

109. Calculate this operation before combining the remaining terms.

     $$\left(-10\right)\times \frac{1}{5}=-2$$

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

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

111. Simplify the imaginary coefficient fraction.

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

112. Combine the reduced real and imaginary terms.

     $$3-2\times i$$

113. Check z and w in both original equations.

     $$(11-i=11-i,i-1=i-1)$$

Answer

The final answer is shown below.