Mathematics 10 — Exercise 3.6 · Q6

Full question

Two game paths satisfy twice the horizontal position plus the vertical position = 5, and horizontal position is one more than vertical. Find their intersection.

Step-by-step solution

1. Let x be horizontal and y vertical: 2x+y=5; x=y+1 gives x−y=1.

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

   $$(\left[\begin{array}{cc}2& 1\\ 1& -1\end{array}\right],\left[\begin{array}{c}x\\ y\end{array}\right],\left[\begin{array}{c}5\\ 1\end{array}\right])$$

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

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

4. Calculate the first diagonal product.

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

5. Calculate the second diagonal product.

   $$1\times 1=1$$

6. Subtract the second product, including its sign.

   $$D=-2-1$$

7. Simplify the subtraction.

   $$D=-3$$

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

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

9. Calculate the first diagonal product.

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

10. Calculate the second diagonal product.

    $$1\times 1=1$$

11. Subtract the second product, including its sign.

    $$D=-2-1$$

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.

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

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

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

16. Calculate this operation before combining the remaining terms.

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

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

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

18. Calculate these two products separately.

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

19. Calculate this operation before combining the remaining terms.

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

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

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

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

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

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

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

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

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

24. Calculate these two products separately.

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

25. Calculate this operation before combining the remaining terms.

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

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

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

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

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

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

29. 2 columns in the first matrix match 2 rows in the second. The product has 2 rows and 1 columns.

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

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

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

31. Calculate this operation before combining the remaining terms.

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

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

    $$\frac{1\times 5}{3\times 1}$$

33. Calculate these two products separately.

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

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

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

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

    $$\mathrm{LCD}=3$$

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

    $$\frac{5}{3}=\frac{5\times 1}{3\times 1}$$

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

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

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

    $$\frac{5+1}{3}=\frac{6}{3}$$

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

    $$\mathrm{GCD}=3$$

40. Show the two divisions explicitly.

    $$\frac{\frac{6}{3}}{\frac{3}{3}}$$

41. Evaluate the numerator division.

    $$\frac{6}{3}=2$$

42. Evaluate the denominator division.

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

43. Write the reduced fraction.

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

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

    $$\frac{1}{3}\times 5+\frac{1}{3}\times 1$$

45. Calculate this individual product, including its signs.

    $$\frac{1}{3}\times 5=\frac{5}{3}$$

46. Calculate this individual product, including its signs.

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

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

    $$\frac{5\times 1}{1\times 3}$$

48. Calculate this operation before combining the remaining terms.

    $$5\times \frac{1}{3}=\frac{5}{3}$$

49. Add the calculated products to obtain this entry.

    $$\frac{5}{3}+\frac{1}{3}=2$$

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

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

51. Calculate these two products separately.

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

52. Calculate this operation before combining the remaining terms.

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

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

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

54. Calculate this operation before combining the remaining terms.

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

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

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

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

    $$\frac{5-2}{3}=\frac{3}{3}$$

57. Show the two divisions explicitly.

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

58. Write the reduced fraction.

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

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

    $$\frac{1}{3}\times 5-\frac{2}{3}$$

60. Calculate this individual product, including its signs.

    $$\frac{1}{3}\times 5=\frac{5}{3}$$

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

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

62. Calculate this operation before combining the remaining terms.

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

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

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

64. Calculate this individual product, including its signs.

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

65. Add the calculated products to obtain this entry.

    $$\frac{5}{3}-\frac{2}{3}=1$$

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

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

67. Calculate this operation before combining the remaining terms.

    $$2\times 2=4$$

68. Calculate this operation before combining the remaining terms.

    $$4+1=5$$

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

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

70. Calculate this operation before combining the remaining terms.

    $$1\times 2=2$$

71. Calculate this operation before combining the remaining terms.

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

72. Calculate this operation before combining the remaining terms.

    $$2-1=1$$

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

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

Answer

The final answer is shown below.