Mathematics 10 — Exercise 3.6 · Q7

Full question

Two years ago a man was five times his son’s age. In six years he will be three times his son’s age. Find present ages.

Step-by-step solution

1. Let current ages be x and y. Two years ago: x−2=5(y−2).

   $$x-2=5\times y-10$$

2. Expand to x−2=5y−10, then collect to x−5y=−8.

3. In six years: x+6=3(y+6). Expand x+6=3y+18, giving x−3y=12.

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

   $$(\left[\begin{array}{cc}1& -5\\ 1& -3\end{array}\right],\left[\begin{array}{c}x\\ y\end{array}\right],\left[\begin{array}{c}-8\\ 12\end{array}\right])$$

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

   $$D=1\times \left(-3\right)-\left(-5\right)\times 1$$

6. Calculate the first diagonal product.

   $$1\times \left(-3\right)=-3$$

7. Calculate the second diagonal product.

   $$\left(-5\right)\times 1=-5$$

8. Subtract the second product, including its sign.

   $$D=-3+5$$

9. Simplify the subtraction.

   $$D=2$$

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

    $$D=1\times \left(-3\right)-\left(-5\right)\times 1$$

11. Calculate the first diagonal product.

    $$1\times \left(-3\right)=-3$$

12. Calculate the second diagonal product.

    $$\left(-5\right)\times 1=-5$$

13. Subtract the second product, including its sign.

    $$D=-3+5$$

14. Simplify the subtraction.

    $$D=2$$

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

    $$2\ne 0$$

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

    $$\left[\begin{array}{cc}-3& -5\\ 1& 1\end{array}\right]$$

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

    $$\left[\begin{array}{cc}-3& 5\\ -1& 1\end{array}\right]$$

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

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

19. Calculate these two products separately.

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

20. Calculate this operation before combining the remaining terms.

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

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

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

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

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

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

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

24. Calculate these two products separately.

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

25. Calculate this operation before combining the remaining terms.

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

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

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

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

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

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

    $$\left[\begin{array}{cc}-\frac{3}{2}& \frac{5}{2}\\ -\frac{1}{2}& \frac{1}{2}\end{array}\right]$$

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

30. 2 columns in the first matrix match 2 rows in the second. The product has 2 rows and 1 columns.

    $$(\left[\begin{array}{cc}-\frac{3}{2}& \frac{5}{2}\\ -\frac{1}{2}& \frac{1}{2}\end{array}\right],\left[\begin{array}{c}-8\\ 12\end{array}\right])$$

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

    $$\frac{5\times 1}{1\times 2}$$

32. Calculate these two products separately.

    $$\frac{5}{2}$$

33. Calculate this operation before combining the remaining terms.

    $$5\times \frac{1}{2}=\frac{5}{2}$$

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

    $$\frac{5\times 12}{2\times 1}$$

35. Calculate these two products separately.

    $$\frac{60}{2}$$

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

    $$\mathrm{GCD}=2$$

37. Show the two divisions explicitly.

    $$\frac{\frac{60}{2}}{\frac{2}{2}}$$

38. Evaluate the numerator division.

    $$\frac{60}{2}=30$$

39. Evaluate the denominator division.

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

40. Write the reduced fraction.

    $$30\times \frac{1}{1}$$

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

    $$--12+\frac{5}{2}\times 12$$

42. Calculate this individual product, including its signs.

    $$\left(-\frac{3}{2}\right)\times \left(-8\right)=12$$

43. Calculate this individual product, including its signs.

    $$\frac{5}{2}\times 12=30$$

44. Add the calculated products to obtain this entry.

    $$12+30=42$$

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

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

46. Calculate these two products separately.

    $$\frac{1}{2}$$

47. Calculate this operation before combining the remaining terms.

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

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

    $$\frac{1\times 12}{2\times 1}$$

49. Calculate these two products separately.

    $$\frac{12}{2}$$

50. Show the two divisions explicitly.

    $$\frac{\frac{12}{2}}{\frac{2}{2}}$$

51. Evaluate the numerator division.

    $$\frac{12}{2}=6$$

52. Write the reduced fraction.

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

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

    $$--4+\frac{1}{2}\times 12$$

54. Calculate this individual product, including its signs.

    $$\left(-\frac{1}{2}\right)\times \left(-8\right)=4$$

55. Calculate this individual product, including its signs.

    $$\frac{1}{2}\times 12=6$$

56. Add the calculated products to obtain this entry.

    $$4+6=10$$

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

    $$\left[\begin{array}{c}42\\ 10\end{array}\right]$$

58. Calculate this operation before combining the remaining terms.

    $$1\times 42=42$$

59. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 50=-50$$

60. Calculate this operation before combining the remaining terms.

    $$42-50=-8$$

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

    $$1\times 42-50=-8$$

62. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 30=-30$$

63. Calculate this operation before combining the remaining terms.

    $$42-30=12$$

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

    $$1\times 42-30=12$$

Answer

The final answer is shown below.