Mathematics 10 — Exercise 1.2 · Q7

Full question

Find the real values of a and b that satisfy the following equation.

Step-by-step solution

1. Expand the brackets and replace i² by −1.

   $$3\times a\times i+a+b\times i-3\times b=11\times i-8$$

2. Equal complex numbers have equal real parts and equal coefficients of i. Match these separately.

   $$(a-3\times b=-8,3\times a+b=11)$$

3. Eliminate the second variable by multiplying the first equation by its coefficient in the second, and the second equation by its coefficient in the first.

   $$(a-3\times b=-8,-9\times a-3\times b=-33)$$

4. Subtract these equations. The second-variable terms cancel.

   $$10\times a=25$$

5. Divide by the remaining coefficient to find the first variable.

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

6. Substitute the first value into an equation containing the second variable.

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

7. Calculate the known product.

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

8. Subtract the known term from both sides.

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

9. Simplify the subtraction.

   $$-3\times b=-\frac{21}{2}$$

10. Divide by the coefficient.

    $$b=-\frac{21}{-6}$$

11. Simplify this fraction.

    $$b=\frac{7}{2}$$

12. Check both values in the original complex equation.

    $$11\times i-8=11\times i-8$$

Answer

The final answer is shown below.