Problem 94 - Elementary Algebra

Question

Perform the operation. $$ (6-7 i)-(12-3 i) $$

Step-by-Step Solution

Answer

The result is \(-6 - 4i\).

Step 1: Identify the expression

The given expression is \((6-7i)-(12-3i)\). We need to perform the subtraction of these two complex numbers.

Step 2: Apply the subtraction rule

Subtract real parts and imaginary parts separately: Real part: \(6 - 12\) and Imaginary part: \(-7i - (-3i)\).

Step 3: Calculate the real part result

Subtract the real parts: \(6 - 12 = -6\).

Step 4: Calculate the imaginary part result

Subtract the imaginary parts: \(-7i - (-3i) = -7i + 3i = -4i\).

Step 5: Combine the results

Combine the results of the real and imaginary parts to get the final result: \(-6 - 4i\).

Key Concepts

Subtraction of Complex Numbers

Complex numbers are composed of two parts: a real part and an imaginary part. In order to subtract complex numbers, you need to handle these parts separately and then combine them back together. Consider the expression

• \( (6-7i) - (12-3i) \)

The subtraction process involves two main steps. First, you will subtract the real parts of the complex numbers. Then, you will subtract the imaginary parts. This ensures each component is dealt with correctly, allowing for easy recombination at the end.

Real and Imaginary Parts

In any complex number, there are two main components:

• The real part: this is the number that does not involve the imaginary unit \(i\).
• The imaginary part: this is the coefficient of \(i\), where \(i\) stands for \(\sqrt{-1}\).

For example, in the complex number \(6 - 7i\), 6 is the real part while \(-7i\) is the imaginary part. Similarly, in \(12 - 3i\), 12 is the real part, and \(-3i\) is the imaginary part.
To subtract these, compute the difference of the corresponding parts:

• Real parts: \(6 - 12 = -6\)
• Imaginary parts: \(-7i - (-3i) = -7i + 3i = -4i\)

The separate treatment of real and imaginary parts is what maintains the integrity of complex number subtraction.

Step by Step Solution

Solving this problem requires a structured approach. Starting with the expression \((6-7i)-(12-3i)\), follow these steps:
1. **Subtract Real Parts:** Take the real components of the complex numbers and subtract one from the other. - Real part calculation: \(6 - 12 = -6\). 2. **Subtract Imaginary Parts:** Take the imaginary components and subtract, being careful about signs: - Imaginary part calculation: \(-7i - (-3i) = -7i + 3i \), which results in \(-4i\).
3. **Combine Results:** Once you have the subtracted values of both parts, your new complex number is: - Final result: \(-6 - 4i\).By breaking down the problem in this way, complex number operations become reachable and comprehensible, allowing for accurate and clear outcomes.