Problem 70 - Algebra 2

Question

Find the values of \(m\) and \(n\) that make each equation true. $$ (m+2 n)+(2 m-n) i=5+5 i $$

Step-by-Step Solution

Answer

The values are \(m = 3\) and \(n = 1\).

Step 1: Identify Real and Imaginary Parts

The given equation is \((m+2n) + (2m-n)i = 5 + 5i\). We start by identifying and separating the real and imaginary parts. The real part is \((m + 2n)\) and the imaginary part is \((2m - n)\).

Step 2: Set Real Parts Equal

Since the equation \((m+2n) + (2m-n)i = 5 + 5i\) must hold true, the real parts must be equal. Thus, equate \(m + 2n = 5\).

Step 3: Set Imaginary Parts Equal

Similarly, for the equation to be true, the imaginary parts must also be equal. Equate \(2m - n = 5\).

Step 4: Solve the System of Equations

We now solve the system of equations: \(m + 2n = 5\) and \(2m - n = 5\). Start by solving the first equation for \(m\):\[ m = 5 - 2n \]

Step 5: Substitute to Solve for m and n

Substitute \(m = 5 - 2n\) into the second equation \(2m - n = 5\):\[ 2(5 - 2n) - n = 5 \]\[ 10 - 4n - n = 5 \]\[ 10 - 5n = 5 \]\[ 5n = 5 \]\[ n = 1 \]

Step 6: Find m Using n

Use \(n = 1\) in the equation \(m = 5 - 2n\) to find \(m\):\[ m = 5 - 2(1) \]\[ m = 3 \]

Step 7: Verification

To verify, substitute \(m = 3\) and \(n = 1\) back into both parts to ensure they satisfy the original equation. The left side becomes \((3+2(1)) + (2 \cdot 3 - 1)i = 5 + 5i\), which matches \(5 + 5i\).

Key Concepts

Real and Imaginary Parts

In complex numbers, each number has two components: the real part and the imaginary part. Consider the equation \((m + 2n) + (2m - n)i = 5 + 5i\). Here, \(m + 2n\) represents the **real part** of the complex number, while \(2m - n\) is the **imaginary part**. Complex numbers are usually written in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part (along with the imaginary unit \(i\), which is the square root of -1).
Understanding how to identify these parts is crucial because it allows us to manipulate and solve equations involving complex numbers. By equating the real parts and the imaginary parts of both sides of the equation separately, you can simplify and solve such problems effectively.

System of Equations

When solving problems involving complex numbers, you often encounter systems of equations. These systems result from setting the real parts and imaginary parts equal separately. In the given equation, after equating, we get two equations: \(m + 2n = 5\) and \(2m - n = 5\).

• The first equation determines a relationship between \(m\) and \(n\) for the real parts to be equal.
• The second equation does the same for the imaginary parts.

This forms a simple linear system of equations that can be solved simultaneously. The beauty of systems of equations is that they allow us to find values for unknowns by solving just enough equations simultaneously. It's a methodical approach that can be applied to a wide range of mathematical and real-world problems.

Unknown Variables

Variables like \(m\) and \(n\) are called **unknown variables** because their values are not initially known. In our complex number equation, the aim is to find values for these variables that satisfy both parts of the equation.
To do so, you start by expressing one variable in terms of the other, as we did by setting \(m = 5 - 2n\) from the real part equation \(m + 2n = 5\). Then, substitute this expression into the second equation (for the imaginary parts), resulting in an equation with just one unknown. Solving this equation will give you the value for one of the variables.
Subsequently, substituting back will provide the value for the other variable. This step-by-step strategy lets you tackle linear equations systematically, ultimately revealing the unknowns.

Verification of Solutions

Once you solve for the unknown variables, it's essential to check your solutions. Verification ensures that the values you found truly satisfy the original equation. In our example, substituting \(m=3\) and \(n=1\) back into both parts of the equation is necessary.

• For the real part: \(m + 2n = 3 + 2(1) = 5\), which matches the original.
• For the imaginary part: \(2m - n = 2(3) - 1 = 5\), confirming the original as well.

Verification acts as a safeguard, protecting against calculation errors and confirming the accuracy of your solutions. It's a fundamental step in problem-solving, ensuring your answers are correct and that you've understood the problem deeply.