When performing addition and multiplication with constant numbers, we treat i as we would treat any variable x in an expression, with the same rules and properties of addition and multiplication.
1)
[tex]( 7 + 5i) - ( -9 + i)=7+5i+9-i=(7+9)+(5i-i)=16+4i[/tex]
2)
Distribute 7+7i over 6 and 7i:
[tex]( 7 + 7i)( 6 + 7i)=( 7 + 7i)\cdot6+( 7 + 7i)\cdot7i=42+42i+49i+49i^2[/tex]
collecting similar terms, and substituting [tex]i^2[/tex] with -1 we have:
42+91i-49=-7+91 i
3)
The conjugate of a complex number a+bi is a-bi;
the conjugate of 1+3i is 1-3i.
Thus, using the difference of squares formula we have
[tex](1+3i)(1-3i)=1^2-(3i)^2=1-9i^2=1-9(-1)=1+9=10[/tex]
4)
To write a rational expression ( with a complex number in the denominator) in the standard form, we multiply by the conjugate of the denominator both the numerator and the denominator:
[tex]\displaystyle{ \frac{3}{3-12i}= \frac{3(3+12i)}{(3-12i)(3+12i)}= \frac{9+36i}{3^2-(12i)^2} [/tex]
[tex]=\displaystyle{ \frac{9+36i}{9+144}= \frac{9+36i}{153}= \frac{9}{153}+ \frac{36}{153}i [/tex]
simplifying by 9, the complex number is finally written as
[tex]\displaystyle{ \frac{1}{17}+ \frac{4}{17}i [/tex]
5)
2 complex numbers a+bi, and c+di are equal only if a=c, and b=d. (Where a, b, c, d are real numbers.)
Thus, x=5, and y=3 is the solution to the equation.
Answers:
1) A
2) A
3) B
4) D
5) (x, y) =( 5, 3)
