Mathematics 10 — Exercise 1.4 · Q2 (iv)

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.

   $$(w+z=3\times i,3\times w+2\times z=2)$$

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

   $$(3\times w+3\times z=9\times i,3\times w+2\times z=2)$$

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

   $$z=9\times i-2$$

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

   $$\frac{1}{1}\times \left(9\times i-2\right)$$

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

   $$1$$

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

   $$\left(\frac{1}{1}\right)\times \left(\frac{1}{1}\times \left(9\times i-2\right)\right)$$

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

   $$\frac{1\times \left(9\times i-2\right)}{1\times 1}$$

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.

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

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

    $$-2-0\times i+9\times 1\times i+9\times 0\times ii$$

11. Real × real: multiply the numerical coefficients.

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

12. Real × imaginary: multiply the numerical coefficients.

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

13. Imaginary × real: multiply the numerical coefficients.

    $$9\times 1\times i=9\times i$$

14. Imaginary × imaginary: multiply the numerical coefficients.

    $$9\times 0\times ii=0\times i^2$$

15. Substitute these four products into the expansion.

    $$-2+0\times i+9\times i+0\times i^2$$

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.

    $$-2+0\times i+9\times i+0$$

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

    $$-2+0+i\times \left(0+9\right)$$

20. Calculate the sum of the real terms.

    $$-2+0=-2$$

21. Calculate the sum of the imaginary coefficients.

    $$0+9=9$$

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

    $$9\times i-2$$

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.

    $$1\times 1+0\times 1\times i$$

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

    $$1\times 1+0\times i+0\times 1\times i+0\times 0\times ii$$

26. Real × real: multiply the numerical coefficients.

    $$1\times 1=1$$

27. Real × imaginary: multiply the numerical coefficients.

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

28. Imaginary × real: multiply the numerical coefficients.

    $$0\times 1\times i=0\times i$$

29. Imaginary × imaginary: multiply the numerical coefficients.

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

30. Substitute these four products into the expansion.

    $$1+0\times i+0\times i+0\times i^2$$

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.

    $$1+0\times i+0\times i+0$$

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

    $$1+0+i\times \left(0+0\right)$$

35. Calculate the sum of the real terms.

    $$1+0=1$$

36. Calculate the sum of the imaginary coefficients.

    $$0+0=0$$

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

    $$0\times i+1$$

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

    $$1=1$$

39. Put the calculated numerator above the calculated denominator.

    $$\frac{1}{1}\times \left(9\times i-2\right)$$

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

    $$-2\times \frac{1}{1}+\frac{1}{1}\times \left(9\times i\right)$$

41. Move i outside its numerical fraction.

    $$-2\times \frac{1}{1}+\left(9\times \frac{1}{1}\right)\times i$$

42. Calculate this operation before combining the remaining terms.

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

43. Simplify the real coefficient fraction.

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

44. Simplify the imaginary coefficient fraction.

    $$9\times \frac{1}{1}=9$$

45. Combine the reduced real and imaginary terms.

    $$9\times i-2$$

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

    $$9\times i+w-2=3\times i$$

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

    $$w=2-6\times i$$

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

    $$\frac{1}{1}\times \left(2-6\times i\right)$$

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

    $$1$$

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

    $$\left(\frac{1}{1}\right)\times \left(\frac{1}{1}\times \left(2-6\times i\right)\right)$$

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

    $$\frac{1\times \left(2-6\times i\right)}{1\times 1}$$

52. Calculate the numerator first.

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

    $$2\times 1-6\times 1\times i$$

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

    $$2\times 1+2\times 0\times i-6\times 1\times i-6\times 0\times ii$$

55. Real × real: multiply the numerical coefficients.

    $$2\times 1=2$$

56. Real × imaginary: multiply the numerical coefficients.

    $$2\times 0\times i=0\times i$$

57. Imaginary × real: multiply the numerical coefficients.

    $$-6\times 1\times i=-6\times i$$

58. Imaginary × imaginary: multiply the numerical coefficients.

    $$-6\times 0\times ii=0\times i^2$$

59. Substitute these four products into the expansion.

    $$2+0\times i-6\times i+0\times i^2$$

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

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

61. Multiply its coefficient by −1.

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

62. Rewrite the expansion after this sign calculation.

    $$2+0\times i-6\times i+0$$

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

    $$2+0+i\times \left(0-6\right)$$

64. Calculate the sum of the real terms.

    $$2+0=2$$

65. Calculate the sum of the imaginary coefficients.

    $$0-6=-6$$

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

    $$2-6\times i$$

67. Now calculate the denominator independently.

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

    $$1\times 1+0\times 1\times i$$

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

    $$1\times 1+0\times i+0\times 1\times i+0\times 0\times ii$$

70. Real × real: multiply the numerical coefficients.

    $$1\times 1=1$$

71. Real × imaginary: multiply the numerical coefficients.

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

72. Imaginary × real: multiply the numerical coefficients.

    $$0\times 1\times i=0\times i$$

73. Imaginary × imaginary: multiply the numerical coefficients.

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

74. Substitute these four products into the expansion.

    $$1+0\times i+0\times i+0\times i^2$$

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

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

76. Multiply its coefficient by −1.

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

77. Rewrite the expansion after this sign calculation.

    $$1+0\times i+0\times i+0$$

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

    $$1+0+i\times \left(0+0\right)$$

79. Calculate the sum of the real terms.

    $$1+0=1$$

80. Calculate the sum of the imaginary coefficients.

    $$0+0=0$$

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

    $$0\times i+1$$

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

    $$1=1$$

83. Put the calculated numerator above the calculated denominator.

    $$\frac{1}{1}\times \left(2-6\times i\right)$$

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

    $$2\times \frac{1}{1}+\frac{1}{1}\times \left(-6\times i\right)$$

85. Move i outside its numerical fraction.

    $$2\times \frac{1}{1}+\left(-6\times \frac{1}{1}\right)\times i$$

86. Simplify the real coefficient fraction.

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

87. Calculate this operation before combining the remaining terms.

    $$\left(-6\right)\times 1=-6$$

88. Simplify the imaginary coefficient fraction.

    $$-6\times \frac{1}{1}=-6$$

89. Combine the reduced real and imaginary terms.

    $$2-6\times i$$

90. Check z and w in both original equations.

    $$(3\times i=3\times i,2=2)$$

Answer

The final answer is shown below.