In an electrical circuits, the voltage, E, in volts, the current, I, in amps, and the opposition to the flow of current, called impedance, Z, in ohms, are related by the equation, `E=IZ`. What is the impedance, in ohms, of an electrical circuit that has a current of (3+i) amps and a voltage of `(-7+i)` volts?

Options

A. `-2+i`
B. `1-2i`
C. `(-11)/(25)-(1)/(5)i`
D. `-(16)/(25)i`

Answer

To find the impedance \( Z \) in ohms of the electrical circuit given the current \( I = 3 + i \) amps and the voltage \( E = -7 + i \) volts, we can use the relationship \( E = IZ \). Rearranging this gives us:

\[ Z = \frac{E}{I} \]

### Step 1: Substitute the values of \( E \) and \( I \)

Substituting the given values into the equation:

\[ Z = \frac{-7 + i}{3 + i} \]

### Step 2: Rationalize the denominator

To simplify this expression, we need to rationalize the denominator. We do this by multiplying both the numerator and the denominator by the conjugate of the denominator, which is \( 3 - i \):

\[ Z = \frac{(-7 + i)(3 - i)}{(3 + i)(3 - i)} \]

### Step 3: Calculate the denominator

Calculating the denominator:

\[ (3 + i)(3 - i) = 3^2 - i^2 = 9 - (-1) = 9 + 1 = 10 \]

### Step 4: Calculate the numerator

Now, calculating the numerator:

\[ (-7 + i)(3 - i) = -21 + 7i + 3i - i^2 \]
Since \( i^2 = -1 \), we have:

\[ -21 + 10i + 1 = -20 + 10i \]

### Step 5: Combine the results

Now we can combine the results:

\[ Z = \frac{-20 + 10i}{10} \]

### Step 6: Simplify the expression

Dividing each term in the numerator by 10 gives:

\[ Z = -2 + i \]

Thus, the impedance \( Z \) of the electrical circuit is:

\[ \boxed{-2 + i} \]

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