Problem 49 - Algebra 2

Question

Find the values of \(m\) and \(n\) that make each equation true. $$ (4+n)+(3 m-7) i=8-2 i $$

Step-by-Step Solution

Answer

Values are \(n = 4\) and \(m = \frac{5}{3}\).

Step 1: Identify Real and Imaginary Parts

To solve the equation \((4+n)+(3m-7)i=8-2i\), we need to separate the real and imaginary parts. The real part of the left side is \(4 + n\) and the imaginary part is \(3m - 7\). The real part of the right side is \(8\) and the imaginary part is \(-2\).

Step 2: Set Real Parts Equal

Since the equation equates complex numbers, their real parts must be equal: \(4 + n = 8\). Solve this equation to find \(n\).

Step 3: Solve for n

\(4 + n = 8\) is simplified to \(n = 8 - 4\). Thus, \(n = 4\).

Step 4: Set Imaginary Parts Equal

The imaginary parts of the equation must also be equal: \(3m - 7 = -2\). Solve this equation to find \(m\).

Step 5: Solve for m

\(3m - 7 = -2\) is simplified to \(3m = -2 + 7\). Therefore, \(3m = 5\), and \(m = \frac{5}{3}\).

Step 6: Verify the Solution

Substitute \(n = 4\) and \(m = \frac{5}{3}\) back into the original equation to ensure both the real and imaginary parts match the right side. The real part becomes \(4 + 4 = 8\) and the imaginary part becomes \(3(\frac{5}{3})-7 = -2\). Both match, so the solution is verified.

Key Concepts

Real and Imaginary Parts

To understand complex numbers, it's crucial to grasp the concepts of real and imaginary parts. A complex number is typically expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The imaginary unit \(i\) is defined as \(\sqrt{-1}\).
This expression allows us to handle complex numbers in a manner similar to algebra with real numbers.When dealing with complex equations like in our problem, it’s essential to separate and compare the real and imaginary parts from both sides of the equation. For example, in our given problem, the expression \[(4+n)+(3m-7)i = 8-2i\]breaks down into:

• Real Part: \(4 + n\) on the left, and \(8\) on the right.
• Imaginary Part: \(3m - 7\) on the left, and \(-2\) on the right.

By separating these components, we can solve for unknowns independently.

Equation Solving

Solving equations involving complex numbers involves setting equal the corresponding real and imaginary components. This technique allows us to simplify complex expressions into standard algebraic equations.
For instance, when given \[4 + n = 8\]and \[3m - 7 = -2\]we can tackle each one separately. To solve the real part equation \(4 + n = 8\), isolate the variable \(n\):

• Subtract 4 from both sides: \(n = 8 - 4\)
• Thus, \(n = 4\)

Similarly, for the imaginary part \(3m - 7 = -2\), follow these steps:

• Add 7 to both sides: \(3m = -2 + 7\)
• Simplify to \(3m = 5\)
• Divide by 3: \(m = \frac{5}{3}\)

Each equation is solved independently but contributes to the overall solution for the complex number.

Algebraic Manipulation

Algebraic manipulation is the crux of solving equations involving complex numbers. Whether dealing with real numbers or complex expressions, algebraic techniques such as simplification, addition, subtraction, and isolation of variables come into play.
In the context of our problem, algebraic manipulation began by isolating the variables \(n\) and \(m\). We simplified them by:

• Separating equations by their real and imaginary components.
• Applying arithmetic operations to solve these separate equations.

Once we've determined the values \(n = 4\) and \(m = \frac{5}{3}\), verification is an important step. Substitute these numbers back into the original equation to ensure both sides become identical:

• The real section becomes \(4 + 4 = 8\)
• The imaginary section becomes \(3(\frac{5}{3}) - 7 = -2\)

Verifying your solution ensures correctness and reliability, showcasing the power of algebraic manipulation in confirming results.