Mathematics 10 — Exercise 6.2 · Q4

Full question

If a+2b=c, find |b|.

Step-by-step solution

1. The orders match. Combine only entries in the same row and column.

   $$(\left[\begin{array}{c}2\\ 1\end{array}\right],\left[\begin{array}{c}1\\ 3\end{array}\right])$$

2. Entry (1,1): subtract the corresponding entries.

   $$2-1=1$$

3. Entry (2,1): subtract the corresponding entries.

   $$1-3=-2$$

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

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

5. Multiply every matrix entry by the scalar.

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

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

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

7. Calculate these two products separately.

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

8. Calculate this operation before combining the remaining terms.

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

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

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

10. Entry (1,1): multiply its value by the scalar.

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

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

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

12. Calculate these two products separately.

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

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

    $$\mathrm{GCD}=2$$

14. Show the two divisions explicitly.

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

15. Evaluate the numerator division.

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

16. Evaluate the denominator division.

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

17. Write the reduced fraction.

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

18. Entry (2,1): multiply its value by the scalar.

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

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

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

20. The first component multiplies î, the second multiplies ĵ.

    $$\frac{î}{2}-ĵ$$

21. Use the horizontal and vertical components as the two perpendicular sides.

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

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

    $$\mathrm{LCD}=4$$

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

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

24. Calculate the converted numerator; the denominator is now common.

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

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

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

26. Calculate the converted numerator; the denominator is now common.

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

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

    $$\frac{1+4}{4}=\frac{5}{4}$$

28. Magnitude is the nonnegative square root of the sum of component squares.

    $$\sqrt{{\left(\frac{1}{2}\right)}^2+{\left(-1\right)}^2}$$

29. Square the horizontal component, including its sign.

    $${\left(\frac{1}{2}\right)}^2=\frac{1}{4}$$

30. Square the vertical component, including its sign.

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

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

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

32. Calculate this operation before combining the remaining terms.

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

33. Add these two squares.

    $$\frac{1}{4}+1=\frac{5}{4}$$

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

    $$\frac{5\times 1}{1\times 4}$$

35. Calculate these two products separately.

    $$\frac{5}{4}$$

36. Calculate this operation before combining the remaining terms.

    $$5\times \frac{1}{4}=\frac{5}{4}$$

37. Take and simplify the positive square root.

    $$\sqrt{\frac{5}{4}}=\frac{1}{2}\times \sqrt{5}$$

Answer

The final answer is shown below.