Mathematics 10 — Exercise 3.6 · Q3

Full question

The branch currents x and y satisfy x+y=7 and 2x−y=2. Find both currents using matrices.

Step-by-step solution

1. Write the coefficients in A, the unknowns in X, and constants in B, giving AX = B.

   $$(\left[\begin{array}{cc}1& 1\\ 2& -1\end{array}\right],\left[\begin{array}{c}x\\ y\end{array}\right],\left[\begin{array}{c}7\\ 2\end{array}\right])$$

2. For a 2 × 2 determinant, multiply the main diagonal and subtract the other diagonal product.

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

3. Calculate the first diagonal product.

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

4. Calculate the second diagonal product.

   $$1\times 2=2$$

5. Subtract the second product, including its sign.

   $$D=-1-2$$

6. Simplify the subtraction.

   $$D=-3$$

7. For a 2 × 2 determinant, multiply the main diagonal and subtract the other diagonal product.

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

8. Calculate the first diagonal product.

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

9. Calculate the second diagonal product.

   $$1\times 2=2$$

10. Subtract the second product, including its sign.

    $$D=-1-2$$

11. Simplify the subtraction.

    $$D=-3$$

12. The determinant is nonzero, so the inverse exists.

    $$-3\ne 0$$

13. Swap the two diagonal entries to begin the adjoint.

    $$\left[\begin{array}{cc}-1& 1\\ 2& 1\end{array}\right]$$

14. Change the signs of the two off-diagonal entries.

    $$\left[\begin{array}{cc}-1& -1\\ -2& 1\end{array}\right]$$

15. Calculate this operation before combining the remaining terms.

    $$\frac{1}{-3}=-\frac{1}{3}$$

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

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

17. Calculate these two products separately.

    $$\frac{1}{3}$$

18. Calculate this operation before combining the remaining terms.

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

19. Divide adjoint entry (1,1) by the determinant.

    $$-\frac{1}{-3}=\frac{1}{3}$$

20. Divide adjoint entry (1,2) by the determinant.

    $$-\frac{1}{-3}=\frac{1}{3}$$

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

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

22. Calculate these two products separately.

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

23. Calculate this operation before combining the remaining terms.

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

24. Divide adjoint entry (2,1) by the determinant.

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

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

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

26. Calculate these two products separately.

    $$-\frac{1}{3}$$

27. Calculate this operation before combining the remaining terms.

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

28. Divide adjoint entry (2,2) by the determinant.

    $$\frac{1}{-3}=-\frac{1}{3}$$

29. Place the calculated entries in their original row and column positions.

    $$\left[\begin{array}{cc}\frac{1}{3}& \frac{1}{3}\\ \frac{2}{3}& -\frac{1}{3}\end{array}\right]$$

30. Premultiply AX = B by A⁻¹. Since A⁻¹A = I, X = A⁻¹B.

31. 2 columns in the first matrix match 2 rows in the second. The product has 2 rows and 1 columns.

    $$(\left[\begin{array}{cc}\frac{1}{3}& \frac{1}{3}\\ \frac{2}{3}& -\frac{1}{3}\end{array}\right],\left[\begin{array}{c}7\\ 2\end{array}\right])$$

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

    $$\frac{1\times 1}{1\times 3}$$

33. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{3}=\frac{1}{3}$$

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

    $$\frac{1\times 7}{3\times 1}$$

35. Calculate these two products separately.

    $$\frac{7}{3}$$

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

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

37. To add these fractions, first use a common denominator: the least common multiple of their denominators.

    $$\mathrm{LCD}=3$$

38. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{7}{3}=\frac{7\times 1}{3\times 1}$$

39. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{2}{3}=\frac{2\times 1}{3\times 1}$$

40. Keep the common denominator and add the signed numerators.

    $$\frac{7+2}{3}=\frac{9}{3}$$

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

    $$\mathrm{GCD}=3$$

42. Show the two divisions explicitly.

    $$\frac{\frac{9}{3}}{\frac{3}{3}}$$

43. Evaluate the numerator division.

    $$\frac{9}{3}=3$$

44. Evaluate the denominator division.

    $$\frac{3}{3}=1$$

45. Write the reduced fraction.

    $$3\times \frac{1}{1}$$

46. Entry (1,1): pair row 1 with column 1 and add their products.

    $$\frac{1}{3}\times 7+\frac{1}{3}\times 2$$

47. Calculate this individual product, including its signs.

    $$\frac{1}{3}\times 7=\frac{7}{3}$$

48. Calculate this individual product, including its signs.

    $$\frac{1}{3}\times 2=\frac{2}{3}$$

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

    $$\frac{7\times 1}{1\times 3}$$

50. Calculate this operation before combining the remaining terms.

    $$7\times \frac{1}{3}=\frac{7}{3}$$

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

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

52. Calculate this operation before combining the remaining terms.

    $$2\times \frac{1}{3}=\frac{2}{3}$$

53. Add the calculated products to obtain this entry.

    $$\frac{7}{3}+\frac{2}{3}=3$$

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

    $$\frac{2\times 7}{3\times 1}$$

55. Calculate these two products separately.

    $$\frac{14}{3}$$

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

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

57. Calculate these two products separately.

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

58. Calculate this operation before combining the remaining terms.

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

59. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{14}{3}=\frac{14\times 1}{3\times 1}$$

60. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

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

61. Keep the common denominator and add the signed numerators.

    $$\frac{14-2}{3}=\frac{12}{3}$$

62. Show the two divisions explicitly.

    $$\frac{\frac{12}{3}}{\frac{3}{3}}$$

63. Evaluate the numerator division.

    $$\frac{12}{3}=4$$

64. Write the reduced fraction.

    $$4\times \frac{1}{1}$$

65. Entry (2,1): pair row 2 with column 1 and add their products.

    $$\frac{2}{3}\times 7-\frac{2}{3}$$

66. Calculate this individual product, including its signs.

    $$\frac{2}{3}\times 7=\frac{14}{3}$$

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

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

68. Calculate this operation before combining the remaining terms.

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

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

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

70. Calculate this individual product, including its signs.

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

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

    $$\frac{14\times 1}{1\times 3}$$

72. Calculate this operation before combining the remaining terms.

    $$14\times \frac{1}{3}=\frac{14}{3}$$

73. Add the calculated products to obtain this entry.

    $$\frac{14}{3}-\frac{2}{3}=4$$

74. Place the calculated entries in their original row and column positions.

    $$\left[\begin{array}{c}3\\ 4\end{array}\right]$$

75. Calculate this operation before combining the remaining terms.

    $$1\times 3=3$$

76. Calculate this operation before combining the remaining terms.

    $$1\times 4=4$$

77. Calculate this operation before combining the remaining terms.

    $$3+4=7$$

78. Check in this original equation by multiplying the coefficients by the calculated values.

    $$1\times 3+1\times 4=7$$

79. Calculate this operation before combining the remaining terms.

    $$2\times 3=6$$

80. Calculate this operation before combining the remaining terms.

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

81. Calculate this operation before combining the remaining terms.

    $$6-4=2$$

82. Check in this original equation by multiplying the coefficients by the calculated values.

    $$2\times 3-4=2$$

Answer

The final answer is shown below.