Mathematics 10 — Exercise Worked examples · Example 21

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.

   $$(w\times \left(-i-3\right)+5\times z=7-i,2\times i\times w+z\times \left(2-i\right)=4)$$

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

   $$(2\times i\times \left(w\times \left(-i-3\right)+5\times z\right)=2\times i\times \left(7-i\right),\left(-i-3\right)\times \left(2\times i\times w+z\times \left(2-i\right)\right)=-4\times i-12)$$

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

   $$z\times \left(9\times i+7\right)=18\times i+14$$

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

   $$\frac{18\times i+14}{9\times i+7}$$

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

   $$7-9\times i$$

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

   $$\left(\frac{7-9\times i}{7-9\times i}\right)\times \left(\frac{18\times i+14}{9\times i+7}\right)$$

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

   $$\frac{\left(7-9\times i\right)\times \left(18\times i+14\right)}{\left(7-9\times i\right)\times \left(9\times i+7\right)}$$

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.

   $$18\times i\times \left(7-9\times i\right)+14\times \left(7-9\times i\right)$$

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

    $$14\times 7+14\times \left(-1\right)\times 9\times i+18\times 7\times i+18\times \left(-9\right)\times ii$$

11. Real × real: multiply the numerical coefficients.

    $$14\times 7=98$$

12. Real × imaginary: multiply the numerical coefficients.

    $$14\times \left(-1\right)\times 9\times i=-126\times i$$

13. Imaginary × real: multiply the numerical coefficients.

    $$18\times 7\times i=126\times i$$

14. Imaginary × imaginary: multiply the numerical coefficients.

    $$18\times \left(-9\right)\times ii=-162\times i^2$$

15. Substitute these four products into the expansion.

    $$98-126\times i+126\times i-162\times i^2$$

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

    $$-162\times i^2=\left(-162\right)\times \left(-1\right)$$

17. Multiply its coefficient by −1.

    $$\left(-162\right)\times \left(-1\right)=162$$

18. Rewrite the expansion after this sign calculation.

    $$98-126\times i+126\times i+162$$

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

    $$98+162+i\times \left(-126+126\right)$$

20. Calculate the sum of the real terms.

    $$98+162=260$$

21. Calculate the sum of the imaginary coefficients.

    $$-126+126=0$$

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

    $$0\times i+260$$

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.

    $$9\times i\times \left(7-9\times i\right)+7\times \left(7-9\times i\right)$$

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

    $$7\times 7+7\times \left(-1\right)\times 9\times i+9\times 7\times i+9\times \left(-9\right)\times ii$$

26. Real × real: multiply the numerical coefficients.

    $$7\times 7=49$$

27. Real × imaginary: multiply the numerical coefficients.

    $$7\times \left(-1\right)\times 9\times i=-63\times i$$

28. Imaginary × real: multiply the numerical coefficients.

    $$9\times 7\times i=63\times i$$

29. Imaginary × imaginary: multiply the numerical coefficients.

    $$9\times \left(-9\right)\times ii=-81\times i^2$$

30. Substitute these four products into the expansion.

    $$49-63\times i+63\times i-81\times i^2$$

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

    $$-81\times i^2=\left(-81\right)\times \left(-1\right)$$

32. Multiply its coefficient by −1.

    $$\left(-81\right)\times \left(-1\right)=81$$

33. Rewrite the expansion after this sign calculation.

    $$49-63\times i+63\times i+81$$

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

    $$49+81+i\times \left(-63+63\right)$$

35. Calculate the sum of the real terms.

    $$49+81=130$$

36. Calculate the sum of the imaginary coefficients.

    $$-63+63=0$$

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

    $$0\times i+130$$

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

    $$130=130$$

39. Put the calculated numerator above the calculated denominator.

    $$\frac{0\times i+260}{130}$$

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

    $$\frac{260}{130}+\frac{0\times i}{130}$$

41. Move i outside its numerical fraction.

    $$\frac{260}{130}+\left(\frac{0}{130}\right)\times i$$

42. Reduce the real coefficient by dividing numerator and denominator by 130.

    $$\frac{260}{130}=\frac{\frac{260}{130}}{\frac{130}{130}}$$

43. Simplify the real coefficient fraction.

    $$\frac{260}{130}=2$$

44. Reduce the imaginary coefficient by dividing numerator and denominator by 130.

    $$\frac{0}{130}=\frac{\frac{0}{130}}{\frac{130}{130}}$$

45. Simplify the imaginary coefficient fraction.

    $$\frac{0}{130}=0$$

46. Combine the reduced real and imaginary terms.

    $$0\times i+2$$

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

    $$w\times \left(-i-3\right)+10=7-i$$

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

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

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

    $$\frac{-i-3}{-i-3}$$

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

    $$i-3$$

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

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

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

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

53. Calculate the numerator first.

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

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

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

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

56. Real × real: multiply the numerical coefficients.

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

57. Real × imaginary: multiply the numerical coefficients.

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

58. Imaginary × real: multiply the numerical coefficients.

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

59. Imaginary × imaginary: multiply the numerical coefficients.

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

60. Substitute these four products into the expansion.

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

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

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

62. Multiply its coefficient by −1.

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

63. Rewrite the expansion after this sign calculation.

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

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

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

65. Calculate the sum of the real terms.

    $$9+1=10$$

66. Calculate the sum of the imaginary coefficients.

    $$-3+3=0$$

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

    $$0\times i+10$$

68. Now calculate the denominator independently.

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

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

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

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

71. Real × real: multiply the numerical coefficients.

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

72. Real × imaginary: multiply the numerical coefficients.

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

73. Imaginary × real: multiply the numerical coefficients.

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

74. Imaginary × imaginary: multiply the numerical coefficients.

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

75. Substitute these four products into the expansion.

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

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

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

77. Multiply its coefficient by −1.

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

78. Rewrite the expansion after this sign calculation.

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

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

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

80. Calculate the sum of the real terms.

    $$9+1=10$$

81. Calculate the sum of the imaginary coefficients.

    $$-3+3=0$$

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

    $$0\times i+10$$

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

    $$10=10$$

84. Put the calculated numerator above the calculated denominator.

    $$\frac{0\times i+10}{10}$$

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

    $$\frac{10}{10}+\frac{0\times i}{10}$$

86. Move i outside its numerical fraction.

    $$\frac{10}{10}+\left(\frac{0}{10}\right)\times i$$

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

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

88. Simplify the real coefficient fraction.

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

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

    $$\frac{0}{10}=\frac{\frac{0}{10}}{\frac{10}{10}}$$

90. Simplify the imaginary coefficient fraction.

    $$\frac{0}{10}=0$$

91. Combine the reduced real and imaginary terms.

    $$0\times i+1$$

92. Check z and w in both original equations.

    $$(7-i=7-i,4=4)$$

Answer

The final answer is shown below.