Mathematics 10 — Exercise 1.1 · Q3 (i)

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.

   $$i\times y-3\times i+2\times x+5=2\times i+1$$

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

   $$(2\times x+5=1,y-3=2)$$

3. Use the equation containing only the first variable.

   $$2\times x+5=1$$

4. Subtract the constant term from both sides.

   $$2\times x+5-5=1-5$$

5. Simplify the subtraction.

   $$2\times x=-4$$

6. Divide both sides by the coefficient.

   $$x=-\frac{4}{2}$$

7. Simplify the fraction.

   $$x=-2$$

8. Use the other original equation to find the second variable.

   $$y-3=2$$

9. Subtract the constant from both sides.

   $$y-3+3=2+3$$

10. Simplify the two sides.

    $$y=5$$

11. Divide both sides by the coefficient.

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

12. Simplify to find the second value.

    $$y=5$$

13. Check both values in the original complex equation.

    $$2\times i+1=2\times i+1$$

Answer

The final answer is shown below.