Problem 13 - Algebra 2

Question

Find the values of \(m\) and \(n\) that make each equation true. $$ 2 m+(3 n+1) i=6-8 i $$

Step-by-Step Solution

Answer

The values are \(m = 3\) and \(n = -3\).

Step 1: Equate Real Parts

The equation given is \(2m + (3n+1)i = 6 - 8i\). First, we equate the real parts of the equation. Since the real part on the right side is 6, we have \(2m = 6\).

Step 2: Solve for m

From \(2m = 6\), divide both sides by 2 to find \(m\). This gives us \(m = 3\).

Step 3: Equate Imaginary Parts

Next, equate the coefficients of the imaginary parts. On the left side, the imaginary part is \(3n + 1\), and on the right side, it is \(-8\). So, we write \(3n + 1 = -8\).

Step 4: Solve for n

To solve \(3n + 1 = -8\), first subtract 1 from both sides to get \(3n = -9\). Then, divide both sides by 3 to find \(n\). This gives us \(n = -3\).

Key Concepts

Understanding Real Part

In complex numbers, each number is composed of a real part and an imaginary part. The real part is simply the component that does not involve the imaginary unit, denoted by \( i \). In the original problem, the equation to solve is \(2m + (3n+1)i = 6 - 8i\). The real part on the left is expressed as \(2m\), while on the right, it is \(6\). To solve for the real portion, we isolate these terms, setting them equal to each other: \(2m = 6\). This allows us to focus solely on the real components without worrying about the imaginary ones at this stage.

By solving this equation, we can deduce the value of \( m \). Since the real part involves only the variable \( m \), every calculation will revolve around real components. Isolating real parts greatly simplifies the problem, turning a complex equation into basic arithmetic.

The Imaginary Part

In complex numbers, the imaginary part contains the imaginary unit \( i \), which is the square root of \(-1\). In our exercise, the imaginary component on the left side is \((3n+1)i\), and on the right, it is \(-8i\). To solve for the imaginary portion, we equate these coefficients: \(3n + 1 = -8\).

This step involves simple algebraic manipulation to solve for \( n \). First, we remove the real constant from the left side, yielding \(3n = -9\), and then divide by 3 to find \( n = -3 \). By focusing entirely on the imaginary components, we separate the problem into manageable parts, solving for the unknowns step by step.

Equation Solving

Solving equations involving complex numbers requires separating the equation into its real and imaginary constituents. This approach allows us to handle each part separately. Complex numbers are algebraic in nature, and this exercise involves linear equations where variables \( m \) and \( n \) are part of real and imaginary terms, respectively.

First, solve the real number equation: \(2m = 6\). Divide both sides by 2, resulting in \(m = 3\).

Next, solve the imaginary number equation: \(3n + 1 = -8\). Simplify this by subtracting 1 from both sides to get \(3n = -9\) and then divide by 3 to find \(n = -3\).

The simplicity of splitting and solving these equations highlights the importance of handling real and imaginary parts separately in complex equations.

Algebraic Manipulation

Algebraic manipulation is key to solving complex number equations. It involves operations like addition, subtraction, multiplication, or division to rearrange equations. In this exercise, after splitting the complex equation into real and imaginary parts, we manipulate each to find unknown values.

For \(2m = 6\), we divide by 2. Similarly, in the imaginary equation \(3n + 1 = -8\), we subtract 1, then divide by 3. These manipulations help in isolating variables to find their values. Manipulating equations in such steps is fundamental to solving problems involving both real and imaginary components.

The process is logical and straightforward as it reduces complex problems to simple arithmetic. This process exemplifies how algebraic manipulation aids in systematically breaking down equations into simpler, solvable parts.