Mathematics 10 — Exercise Review · Q8

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.

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

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

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

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

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

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

   $$\frac{4\times i+2}{i-2}$$

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

   $$-i-2$$

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

   $$\left(\frac{-i-2}{-i-2}\right)\times \left(\frac{4\times i+2}{i-2}\right)$$

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

   $$\frac{\left(-i-2\right)\times \left(4\times i+2\right)}{\left(-i-2\right)\times \left(i-2\right)}$$

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.

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

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

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

11. Real × real: multiply the numerical coefficients.

    $$2\times \left(-2\right)=-4$$

12. Real × imaginary: multiply the numerical coefficients.

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

13. Imaginary × real: multiply the numerical coefficients.

    $$4\times \left(-2\right)\times i=-8\times i$$

14. Imaginary × imaginary: multiply the numerical coefficients.

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

15. Substitute these four products into the expansion.

    $$-4-2\times i-8\times i-4\times i^2$$

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

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

17. Multiply its coefficient by −1.

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

18. Rewrite the expansion after this sign calculation.

    $$-4-2\times i-8\times i+4$$

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

    $$-4+4+i\times \left(-2-8\right)$$

20. Calculate the sum of the real terms.

    $$-4+4=0$$

21. Calculate this operation before combining the remaining terms.

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

22. Calculate the sum of the imaginary coefficients.

    $$-2-8=-10$$

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

    $$0-10\times i$$

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.

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

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

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

27. Real × real: multiply the numerical coefficients.

    $$\left(-2\right)\times \left(-2\right)=4$$

28. Real × imaginary: multiply the numerical coefficients.

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

29. Imaginary × real: multiply the numerical coefficients.

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

30. Imaginary × imaginary: multiply the numerical coefficients.

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

31. Substitute these four products into the expansion.

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

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

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

33. Multiply its coefficient by −1.

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

34. Rewrite the expansion after this sign calculation.

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

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

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

36. Calculate the sum of the real terms.

    $$4+1=5$$

37. Calculate the sum of the imaginary coefficients.

    $$2-2=0$$

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

    $$0\times i+5$$

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

    $$5=5$$

40. Put the calculated numerator above the calculated denominator.

    $$\frac{0-10\times i}{5}$$

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

    $$\frac{0}{5}+\frac{-10\times i}{5}$$

42. Move i outside its numerical fraction.

    $$\frac{0}{5}+\left(-\frac{10}{5}\right)\times i$$

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

    $$\frac{0}{5}=\frac{\frac{0}{5}}{\frac{5}{5}}$$

44. Simplify the real coefficient fraction.

    $$\frac{0}{5}=0$$

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

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

46. Calculate these two products separately.

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

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

    $$\mathrm{GCD}=5$$

48. Show the two divisions explicitly.

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

49. Evaluate the denominator division.

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

50. Write the reduced fraction.

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

51. Calculate this operation before combining the remaining terms.

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

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

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

53. Simplify the imaginary coefficient fraction.

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

54. Combine the reduced real and imaginary terms.

    $$0-2\times i$$

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

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

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

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

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

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

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

    $$i+2$$

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

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

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

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

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.

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

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

    $$2\times 2+2\times 1\times i+2\times i+1\times ii$$

64. Real × real: multiply the numerical coefficients.

    $$2\times 2=4$$

65. Real × imaginary: multiply the numerical coefficients.

    $$2\times 1\times i=2\times i$$

66. Imaginary × real: multiply the numerical coefficients.

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

67. Imaginary × imaginary: multiply the numerical coefficients.

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

68. Substitute these four products into the expansion.

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

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

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

70. Multiply its coefficient by −1.

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

71. Rewrite the expansion after this sign calculation.

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

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

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

73. Calculate the sum of the real terms.

    $$4-1=3$$

74. Calculate the sum of the imaginary coefficients.

    $$2+2=4$$

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

    $$4\times i+3$$

76. Now calculate the denominator independently.

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

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

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

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

79. Real × real: multiply the numerical coefficients.

    $$2\times 2=4$$

80. Real × imaginary: multiply the numerical coefficients.

    $$2\times 1\times i=2\times i$$

81. Imaginary × real: multiply the numerical coefficients.

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

82. Imaginary × imaginary: multiply the numerical coefficients.

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

83. Substitute these four products into the expansion.

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

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

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

85. Multiply its coefficient by −1.

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

86. Rewrite the expansion after this sign calculation.

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

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

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

88. Calculate the sum of the real terms.

    $$4+1=5$$

89. Calculate the sum of the imaginary coefficients.

    $$2-2=0$$

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

    $$0\times i+5$$

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

    $$5=5$$

92. Put the calculated numerator above the calculated denominator.

    $$\frac{4\times i+3}{5}$$

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

    $$\frac{3}{5}+\frac{4\times i}{5}$$

94. Move i outside its numerical fraction.

    $$\frac{3}{5}+\left(\frac{4}{5}\right)\times i$$

95. Simplify the real coefficient fraction.

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

96. Simplify the imaginary coefficient fraction.

    $$\frac{4}{5}=\frac{4}{5}$$

97. Combine the reduced real and imaginary terms.

    $$\frac{4\times i}{5}+\frac{3}{5}$$

98. Check z and w in both original equations.

    $$(i+4=i+4,3\times i+3=3\times i+3)$$

Answer

The final answer is shown below.