Mathematics 10 — Exercise 3.6 · Q2 (i)

Full question

Solve using Cramer’s rule.

Step-by-step solution

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

   $$(\left[\begin{array}{cc}1& 4\\ 2& -1\end{array}\right],\left[\begin{array}{c}x\\ y\end{array}\right],\left[\begin{array}{c}4\\ 5\end{array}\right])$$

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

   $$D=1\times \left(-1\right)-4\times 2$$

3. Calculate the first diagonal product.

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

4. Calculate the second diagonal product.

   $$4\times 2=8$$

5. Subtract the second product, including its sign.

   $$D=-1-8$$

6. Simplify the subtraction.

   $$D=-9$$

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

   $$\left[\begin{array}{cc}4& 4\\ 5& -1\end{array}\right]$$

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

   $$\mathrm{Dₓ}=4\times \left(-1\right)-4\times 5$$

9. Calculate the first diagonal product.

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

10. Calculate the second diagonal product.

    $$4\times 5=20$$

11. Subtract the second product, including its sign.

    $$\mathrm{Dₓ}=-4-20$$

12. Simplify the subtraction.

    $$\mathrm{Dₓ}=-24$$

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

    $$\left[\begin{array}{cc}1& 4\\ 2& 5\end{array}\right]$$

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

    $$\mathrm{Dᵧ}=1\times 5-4\times 2$$

15. Calculate the first diagonal product.

    $$1\times 5=5$$

16. Calculate the second diagonal product.

    $$4\times 2=8$$

17. Subtract the second product, including its sign.

    $$\mathrm{Dᵧ}=5-8$$

18. Simplify the subtraction.

    $$\mathrm{Dᵧ}=-3$$

19. Divide the replacement determinant by D.

    $$x=-\frac{24}{-9}$$

20. Reduce this quotient to obtain the variable value.

    $$x=\frac{8}{3}$$

21. Divide the replacement determinant by D.

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

22. Reduce this quotient to obtain the variable value.

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

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

    $$\frac{8\times 1}{1\times 3}$$

24. Calculate these two products separately.

    $$\frac{8}{3}$$

25. Calculate this operation before combining the remaining terms.

    $$8\times \frac{1}{3}=\frac{8}{3}$$

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

    $$\frac{1\times 8}{1\times 3}$$

27. Calculate this operation before combining the remaining terms.

    $$1\times \frac{8}{3}=\frac{8}{3}$$

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

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

29. Calculate these two products separately.

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

30. Calculate this operation before combining the remaining terms.

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

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

    $$\frac{4\times 1}{1\times 3}$$

32. Calculate these two products separately.

    $$\frac{4}{3}$$

33. Calculate this operation before combining the remaining terms.

    $$4\times \frac{1}{3}=\frac{4}{3}$$

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

    $$\mathrm{LCD}=3$$

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

    $$\frac{8}{3}=\frac{8\times 1}{3\times 1}$$

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

    $$\frac{4}{3}=\frac{4\times 1}{3\times 1}$$

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

    $$\frac{8+4}{3}=\frac{12}{3}$$

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

    $$\mathrm{GCD}=3$$

39. Show the two divisions explicitly.

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

40. Evaluate the numerator division.

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

41. Evaluate the denominator division.

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

42. Write the reduced fraction.

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

43. Calculate this operation before combining the remaining terms.

    $$\frac{8}{3}+\frac{4}{3}=4$$

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

    $$1\times \frac{8}{3}+4\times \frac{1}{3}=4$$

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

    $$\frac{2\times 8}{1\times 3}$$

46. Calculate these two products separately.

    $$\frac{16}{3}$$

47. Calculate this operation before combining the remaining terms.

    $$2\times \frac{8}{3}=\frac{16}{3}$$

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

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

49. Calculate these two products separately.

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

50. Calculate this operation before combining the remaining terms.

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

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

    $$\frac{16}{3}=\frac{16\times 1}{3\times 1}$$

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

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

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

    $$\frac{16-1}{3}=\frac{15}{3}$$

54. Show the two divisions explicitly.

    $$\frac{\frac{15}{3}}{\frac{3}{3}}$$

55. Evaluate the numerator division.

    $$\frac{15}{3}=5$$

56. Write the reduced fraction.

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

57. Calculate this operation before combining the remaining terms.

    $$\frac{16}{3}-\frac{1}{3}=5$$

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

    $$2\times \frac{8}{3}-\frac{1}{3}=5$$

Answer

The final answer is shown below.