Mathematics 10 — Exercise 6.2 · Q6

Full question

Find m and n in this vector equation.

Step-by-step solution

1. Distribute m and n to BOTH components: mî−10mĵ+4nî−3nĵ. Then match î and ĵ coefficients separately.

2. These corresponding entries give a pair of linear equations.

   $$(m+4\times n=5,-10\times m-3\times n=-3)$$

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

   $$(\left[\begin{array}{cc}1& 4\\ -10& -3\end{array}\right],\left[\begin{array}{c}m\\ n\end{array}\right],\left[\begin{array}{c}5\\ -3\end{array}\right])$$

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

   $$D=1\times \left(-3\right)-4\times \left(-10\right)$$

5. Calculate the first diagonal product.

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

6. Calculate the second diagonal product.

   $$4\times \left(-10\right)=-40$$

7. Subtract the second product, including its sign.

   $$D=-3+40$$

8. Simplify the subtraction.

   $$D=37$$

9. For Dₓ replace the first coefficient column by the constants.

   $$\left[\begin{array}{cc}5& 4\\ -3& -3\end{array}\right]$$

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

    $$\mathrm{Dₓ}=5\times \left(-3\right)-4\times \left(-3\right)$$

11. Calculate the first diagonal product.

    $$5\times \left(-3\right)=-15$$

12. Calculate the second diagonal product.

    $$4\times \left(-3\right)=-12$$

13. Subtract the second product, including its sign.

    $$\mathrm{Dₓ}=-15+12$$

14. Simplify the subtraction.

    $$\mathrm{Dₓ}=-3$$

15. For Dᵧ replace the second coefficient column by the constants.

    $$\left[\begin{array}{cc}1& 5\\ -10& -3\end{array}\right]$$

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

    $$\mathrm{Dᵧ}=1\times \left(-3\right)-5\times \left(-10\right)$$

17. Calculate the first diagonal product.

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

18. Calculate the second diagonal product.

    $$5\times \left(-10\right)=-50$$

19. Subtract the second product, including its sign.

    $$\mathrm{Dᵧ}=-3+50$$

20. Simplify the subtraction.

    $$\mathrm{Dᵧ}=47$$

21. Divide the replacement determinant by D.

    $$m=-\frac{3}{37}$$

22. Reduce this quotient to obtain the variable value.

    $$m=-\frac{3}{37}$$

23. Divide the replacement determinant by D.

    $$n=\frac{47}{37}$$

24. Reduce this quotient to obtain the variable value.

    $$n=\frac{47}{37}$$

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

    $$\frac{\left(-3\right)\times 1}{1\times 37}$$

26. Calculate these two products separately.

    $$-\frac{3}{37}$$

27. Calculate this operation before combining the remaining terms.

    $$\left(-3\right)\times \frac{1}{37}=-\frac{3}{37}$$

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

    $$\frac{1\times \left(-3\right)}{1\times 37}$$

29. Calculate this operation before combining the remaining terms.

    $$1\times \left(-\frac{3}{37}\right)=-\frac{3}{37}$$

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

    $$\frac{47\times 1}{1\times 37}$$

31. Calculate these two products separately.

    $$\frac{47}{37}$$

32. Calculate this operation before combining the remaining terms.

    $$47\times \frac{1}{37}=\frac{47}{37}$$

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

    $$\frac{4\times 47}{1\times 37}$$

34. Calculate these two products separately.

    $$\frac{188}{37}$$

35. Calculate this operation before combining the remaining terms.

    $$4\times \frac{47}{37}=\frac{188}{37}$$

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

    $$\mathrm{LCD}=37$$

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

    $$-\frac{3}{37}=\frac{\left(-3\right)\times 1}{37\times 1}$$

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

    $$\frac{188}{37}=\frac{188\times 1}{37\times 1}$$

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

    $$\frac{-3+188}{37}=\frac{185}{37}$$

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

    $$\mathrm{GCD}=37$$

41. Show the two divisions explicitly.

    $$\frac{\frac{185}{37}}{\frac{37}{37}}$$

42. Evaluate the numerator division.

    $$\frac{185}{37}=5$$

43. Evaluate the denominator division.

    $$\frac{37}{37}=1$$

44. Write the reduced fraction.

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

45. Calculate this operation before combining the remaining terms.

    $$-\frac{3}{37}+\frac{188}{37}=5$$

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

    $$1\times \left(-\frac{3}{37}\right)+4\times \frac{47}{37}=5$$

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

    $$\frac{30\times 1}{1\times 37}$$

48. Calculate these two products separately.

    $$\frac{30}{37}$$

49. Calculate this operation before combining the remaining terms.

    $$30\times \frac{1}{37}=\frac{30}{37}$$

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

    $$\frac{141\times 1}{1\times 37}$$

51. Calculate these two products separately.

    $$\frac{141}{37}$$

52. Calculate this operation before combining the remaining terms.

    $$141\times \frac{1}{37}=\frac{141}{37}$$

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

    $$\frac{\left(-1\right)\times 141}{1\times 37}$$

54. Calculate these two products separately.

    $$-\frac{141}{37}$$

55. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{141}{37}=-\frac{141}{37}$$

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

    $$\frac{30}{37}=\frac{30\times 1}{37\times 1}$$

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

    $$-\frac{141}{37}=\frac{\left(-141\right)\times 1}{37\times 1}$$

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

    $$\frac{30-141}{37}=-\frac{111}{37}$$

59. Show the two divisions explicitly.

    $$\frac{-\frac{111}{37}}{\frac{37}{37}}$$

60. Evaluate the numerator division.

    $$-\frac{111}{37}=-3$$

61. Write the reduced fraction.

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

62. Calculate this operation before combining the remaining terms.

    $$\frac{30}{37}-\frac{141}{37}=-3$$

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

    $$-(-\frac{30}{37})-\frac{141}{37}=-3$$

64. Check the values in the original two matrices.

    $$(\left[\begin{array}{c}5\\ -3\end{array}\right],\left[\begin{array}{c}5\\ -3\end{array}\right])$$

Answer

The final answer is shown below.