Mathematics 10 — Exercise 1.1 · Q3 (iv)

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.

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

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

   $$(x+2\times y=4,-x+y=-1)$$

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.

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

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

   $$3\times x=6$$

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

   $$x=2$$

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

   $$1\times 2+2\times y=4$$

7. Calculate the known product.

   $$2\times y+2=4$$

8. Subtract the known term from both sides.

   $$2\times y=4-2$$

9. Simplify the subtraction.

   $$2\times y=2$$

10. Divide by the coefficient.

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

11. Simplify this fraction.

    $$y=1$$

12. Check both values in the original complex equation.

    $$4-i=4-i$$

Answer

The final answer is shown below.