to question in math 115,Need answer

Question

Complex numbers are used to describe​ current, I,​ voltage, E, and​ impedance, Z. These three quantities are related by the equation

E=IZ.

Given two of these​ quantities, solve the equation

E=IZ

for the missing variable.

I=4+3i, E=1+9i

Answer 1

E = IZ

You are given I and E, so they want you to solve for Z. To do that, divide both sides by I (so Z will be alone on the right):

E/I = Z

This can be written more conveniently:

Z = E/I

Substituting in the values given for E and I:

Z = (1+9i)/(4+3i)

Multiply the right hand side by (4-3i)/(4-3i) (which =1 so we're not changing anything):

Z = (1+9i)/(4+3i) * (4-3i)/(4-3i)

Z = ((1+9i)(4-3i)) / ((4+3i)(4-3i))

Simplify:

Z = (4-3i+36i-27i² ) / (16+12i-12i-9i²)

Z = (31+33i) / 25

[ Notice how you can always make the denominator a real number by multiplying by the complex conjugate: (a+bi)(a-bi) = a² -abi + abi - b²i² = a²+b² , making use of i² = -1. ]