Mathematics 10 — Exercise Review · Q6

Full question

Twice the first number plus the second is 21; the first plus twice the second is 27. Find both numbers.

Step-by-step solution

1. Let x be the first number and y the second.

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

   $$(\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right],\left[\begin{array}{c}x\\ y\end{array}\right],\left[\begin{array}{c}21\\ 27\end{array}\right])$$

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

   $$D=2\times 2-1\times 1$$

4. Calculate the first diagonal product.

   $$2\times 2=4$$

5. Calculate the second diagonal product.

   $$1\times 1=1$$

6. Subtract the second product, including its sign.

   $$D=4-1$$

7. Simplify the subtraction.

   $$D=3$$

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

   $$D=2\times 2-1\times 1$$

9. Calculate the first diagonal product.

   $$2\times 2=4$$

10. Calculate the second diagonal product.

    $$1\times 1=1$$

11. Subtract the second product, including its sign.

    $$D=4-1$$

12. Simplify the subtraction.

    $$D=3$$

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

    $$3\ne 0$$

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

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

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

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

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

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

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

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

18. Calculate these two products separately.

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

19. Calculate this operation before combining the remaining terms.

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

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

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

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

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

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

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

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

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

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

25. 2 columns in the first matrix match 2 rows in the second. The product has 2 rows and 1 columns.

    $$(\left[\begin{array}{cc}\frac{2}{3}& -\frac{1}{3}\\ -\frac{1}{3}& \frac{2}{3}\end{array}\right],\left[\begin{array}{c}21\\ 27\end{array}\right])$$

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

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

27. Calculate these two products separately.

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

28. Calculate this operation before combining the remaining terms.

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

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

    $$\frac{2\times 21}{3\times 1}$$

30. Calculate these two products separately.

    $$\frac{42}{3}$$

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

    $$\mathrm{GCD}=3$$

32. Show the two divisions explicitly.

    $$\frac{\frac{42}{3}}{\frac{3}{3}}$$

33. Evaluate the numerator division.

    $$\frac{42}{3}=14$$

34. Evaluate the denominator division.

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

35. Write the reduced fraction.

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

36. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 9=-9$$

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

    $$\frac{2}{3}\times 21-9$$

38. Calculate this individual product, including its signs.

    $$\frac{2}{3}\times 21=14$$

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

    $$\frac{\left(-1\right)\times 27}{3\times 1}$$

40. Calculate these two products separately.

    $$-\frac{27}{3}$$

41. Show the two divisions explicitly.

    $$\frac{-\frac{27}{3}}{\frac{3}{3}}$$

42. Evaluate the numerator division.

    $$-\frac{27}{3}=-9$$

43. Write the reduced fraction.

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

44. Calculate this individual product, including its signs.

    $$\left(-\frac{1}{3}\right)\times 27=-9$$

45. Add the calculated products to obtain this entry.

    $$14-9=5$$

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

    $$\frac{2\times 27}{3\times 1}$$

47. Calculate these two products separately.

    $$\frac{54}{3}$$

48. Show the two divisions explicitly.

    $$\frac{\frac{54}{3}}{\frac{3}{3}}$$

49. Evaluate the numerator division.

    $$\frac{54}{3}=18$$

50. Write the reduced fraction.

    $$18\times \frac{1}{1}$$

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

    $$-7+\frac{2}{3}\times 27$$

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

    $$\frac{\left(-1\right)\times 21}{3\times 1}$$

53. Calculate these two products separately.

    $$-\frac{21}{3}$$

54. Show the two divisions explicitly.

    $$\frac{-\frac{21}{3}}{\frac{3}{3}}$$

55. Evaluate the numerator division.

    $$-\frac{21}{3}=-7$$

56. Write the reduced fraction.

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

57. Calculate this individual product, including its signs.

    $$\left(-\frac{1}{3}\right)\times 21=-7$$

58. Calculate this individual product, including its signs.

    $$\frac{2}{3}\times 27=18$$

59. Add the calculated products to obtain this entry.

    $$-7+18=11$$

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

    $$\left[\begin{array}{c}5\\ 11\end{array}\right]$$

61. Calculate this operation before combining the remaining terms.

    $$2\times 5=10$$

62. Calculate this operation before combining the remaining terms.

    $$1\times 11=11$$

63. Calculate this operation before combining the remaining terms.

    $$10+11=21$$

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

    $$2\times 5+1\times 11=21$$

65. Calculate this operation before combining the remaining terms.

    $$1\times 5=5$$

66. Calculate this operation before combining the remaining terms.

    $$2\times 11=22$$

67. Calculate this operation before combining the remaining terms.

    $$5+22=27$$

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

    $$1\times 5+2\times 11=27$$

Answer

The final answer is shown below.