Problem 38 - Precalculus: Functions and Graphs

Question

Find the values of \(x\) and \(y,\) where \(x\) and \(y\) are real numbers. $$8+(3 x+y) i=2 x-4 i$$

Step-by-Step Solution

Answer

\(x = 4\) and \(y = -16\).

Step 1: Equate Real Parts

To find the values of \(x\) and \(y\), we start by equating the real parts on both sides of the equation. The expression on the left is \(8 + (3x + y)i\), and the right side is \(2x - 4i\). The real part on the left is \(8\), and on the right, the real part is \(2x\). Thus, we have the equation \(8 = 2x\).

Step 2: Solve for x

From the equation \(8 = 2x\), we can solve for \(x\) by dividing both sides by 2: \[ x = \frac{8}{2} = 4. \]

Step 3: Equate Imaginary Parts

Now, we equate the imaginary parts. On the left, the imaginary part is \((3x + y)i\), and on the right, it is \(-4i\). This gives us the equation \(3x + y = -4\).

Step 4: Substitute x Value

Substitute \(x = 4\) into the equation \(3x + y = -4\): \[ 3(4) + y = -4. \] Simplify this to find \(y\): \[ 12 + y = -4. \]

Step 5: Solve for y

To find \(y\), subtract 12 from both sides of the equation: \[ y = -4 - 12 = -16. \]

Key Concepts

Equating Real and Imaginary Parts

Let's dive into understanding complex numbers. A complex number comprises two parts: a real part and an imaginary part. In any equation involving complex numbers, these parts can be handled separately. This property comes in handy when trying to solve equations involving both real and imaginary numbers. Given an equation such as: 8 + (3x + y)i = 2x - 4i, we can separate this into two different parts:

• The real part: 8 = 2x
• The imaginary part: (3x + y)i = -4i

By equating the real and imaginary parts separately, we simplify our problem into manageable linear equations. This approach allows us to find the values of unknowns in a straightforward manner.

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra. Here, we'll use this skill to find the unknowns in our equations. In our exercise, once we've equated the real parts and obtained:8 = 2x,we solve for x by isolating it on one side of the equation. Divide both sides by 2, we find:x = \(\frac{8}{2} = 4\).This kind of linear equation is very simple since it involves a straightforward manipulation. This example shows the importance of understanding the structure of algebraic equations and performing operations systematically. Through reasoning and systematic procedures, we can resolve unknowns effectively.

Substitution Method

The substitution method can be a very effective tool when working with equations involving multiple unknowns. After solving for one variable, we can substitute it into another equation to solve for the next unknown. In our exercise, once we found that: x = 4, we substituted this value into the imaginary part equation: 3x + y = -4. Replacing x with 4 gives: 3(4) + y = -4. Simplify it to solve for y: 12 + y = -4. To isolate y, subtract 12 from both sides: y = -4 - 12 = -16. Substitution allows us to use known values to simplify our equations and find solutions step by step. This method is crucial for managing more complex problems systematically and is especially useful when handling equations with more than one variable.