Mathematics 10 — Exercise 3.6 · Q2 (iv)

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.

   $$(\left[\begin{array}{cc}3& 2\\ 5& 6\end{array}\right],\left[\begin{array}{c}x\\ y\end{array}\right],\left[\begin{array}{c}-1\\ 5\end{array}\right])$$

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

   $$D=3\times 6-2\times 5$$

3. Calculate the first diagonal product.

   $$3\times 6=18$$

4. Calculate the second diagonal product.

   $$2\times 5=10$$

5. Subtract the second product, including its sign.

   $$D=18-10$$

6. Simplify the subtraction.

   $$D=8$$

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

   $$\left[\begin{array}{cc}-1& 2\\ 5& 6\end{array}\right]$$

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

   $$\mathrm{Dₓ}=-6-2\times 5$$

9. Calculate the first diagonal product.

   $$\left(-1\right)\times 6=-6$$

10. Calculate the second diagonal product.

    $$2\times 5=10$$

11. Subtract the second product, including its sign.

    $$\mathrm{Dₓ}=-6-10$$

12. Simplify the subtraction.

    $$\mathrm{Dₓ}=-16$$

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

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

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

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

15. Calculate the first diagonal product.

    $$3\times 5=15$$

16. Calculate the second diagonal product.

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

17. Subtract the second product, including its sign.

    $$\mathrm{Dᵧ}=15+5$$

18. Simplify the subtraction.

    $$\mathrm{Dᵧ}=20$$

19. Divide the replacement determinant by D.

    $$x=-\frac{16}{8}$$

20. Reduce this quotient to obtain the variable value.

    $$x=-2$$

21. Divide the replacement determinant by D.

    $$y=\frac{20}{8}$$

22. Reduce this quotient to obtain the variable value.

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

23. Calculate this operation before combining the remaining terms.

    $$3\times \left(-2\right)=-6$$

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

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

25. Calculate these two products separately.

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

26. Calculate this operation before combining the remaining terms.

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

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

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

28. Calculate these two products separately.

    $$\frac{10}{2}$$

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

    $$\mathrm{GCD}=2$$

30. Show the two divisions explicitly.

    $$\frac{\frac{10}{2}}{\frac{2}{2}}$$

31. Evaluate the numerator division.

    $$\frac{10}{2}=5$$

32. Evaluate the denominator division.

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

33. Write the reduced fraction.

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

34. Calculate this operation before combining the remaining terms.

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

35. Calculate this operation before combining the remaining terms.

    $$-6+5=-1$$

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

    $$3\times \left(-2\right)+2\times \frac{5}{2}=-1$$

37. Calculate this operation before combining the remaining terms.

    $$5\times \left(-2\right)=-10$$

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

    $$\frac{6\times 5}{1\times 2}$$

39. Calculate these two products separately.

    $$\frac{30}{2}$$

40. Show the two divisions explicitly.

    $$\frac{\frac{30}{2}}{\frac{2}{2}}$$

41. Evaluate the numerator division.

    $$\frac{30}{2}=15$$

42. Write the reduced fraction.

    $$15\times \frac{1}{1}$$

43. Calculate this operation before combining the remaining terms.

    $$6\times \frac{5}{2}=15$$

44. Calculate this operation before combining the remaining terms.

    $$-10+15=5$$

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

    $$5\times \left(-2\right)+6\times \frac{5}{2}=5$$

Answer

The final answer is shown below.