Mathematics 10 — Exercise Review · Q9

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.

   $$-4\times a\times i+3\times a+3\times b\times i+4\times b=1$$

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

   $$(3\times a+4\times b=1,-4\times a+3\times b=0)$$

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.

   $$(9\times a+12\times b=3,-16\times a+12\times b=0)$$

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

   $$25\times a=3$$

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

   $$a=\frac{3}{25}$$

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

   $$3\times \frac{3}{25}+4\times b=1$$

7. Calculate the known product.

   $$4\times b+\frac{9}{25}=1$$

8. Subtract the known term from both sides.

   $$4\times b=1-\frac{9}{25}$$

9. Simplify the subtraction.

   $$4\times b=\frac{16}{25}$$

10. Divide by the coefficient.

    $$b=\frac{16}{100}$$

11. Simplify this fraction.

    $$b=\frac{4}{25}$$

12. Check both values in the original complex equation.

    $$1=1$$

Answer

The final answer is shown below.