Mathematics 10 — Exercise 1.2 · Q6

Full question

Find the real values of x and y that satisfy the following equality of complex numbers.

Step-by-step solution

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

   $$-2\times i\times x+i\times y+2\times x+y=2\times i+4$$

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

   $$(2\times x+y=4,-2\times x+y=2)$$

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.

   $$(2\times x+y=4,-2\times x+y=2)$$

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

   $$4\times x=2$$

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

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

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

   $$2\times \frac{1}{2}+y=4$$

7. Calculate the known product.

   $$y+1=4$$

8. Subtract the known term from both sides.

   $$y=4-1$$

9. Simplify the subtraction.

   $$y=3$$

10. Divide by the coefficient.

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

11. Simplify this fraction.

    $$y=3$$

12. Check both values in the original complex equation.

    $$2\times i+4=2\times i+4$$

Answer

The final answer is shown below.