Problem 36 - Algebra and Trigonometry with Analytic Geometry

Question

Find the values of \(x\) and \(y\), where \(x\) and \(y\) are $$ (x-y)+3 i=7+y i $$

Step-by-Step Solution

Answer

The values are \(x = 10\) and \(y = 3\).

Step 1: Equate Real Parts

In the given equation \((x-y) + 3i = 7 + yi\), identify the real parts of the equation. The real part on the left is \(x-y\) and the real part on the right is \(7\). Equating them gives us the equation: \[x-y = 7\]

Step 2: Equate Imaginary Parts

Now, identify the imaginary parts. The imaginary part on the left is \(3i\) and on the right is \(yi\). Equate them to get:\[3 = y\]

Step 3: Solve for y

From Step 2, we already have the value of \(y\). Therefore, \(y = 3\) directly from the equation \(3 = y\).

Step 4: Substitute y and Solve for x

Using the value of \(y\) from Step 3, substitute \(y = 3\) into the equation from Step 1: \[x - 3 = 7\]Add 3 to both sides of the equation to solve for \(x\):\[x = 7 + 3 = 10\]

Key Concepts

Equating Real and Imaginary Parts

When dealing with complex numbers, it’s important to understand the difference between real and imaginary parts. A complex number generally consists of a real part and an imaginary part. For example, in the expression \((x-y) + 3i = 7 + yi\), we see both of these.

Complex numbers are often written in the form \(a + bi\). In this form, \(a\) represents the real part and \(bi\) represents the imaginary part. To solve the equation given, we first separate and equate these parts.

• Real Parts: Look at values without the imaginary unit \(i\). For the given equation, the real part of the left side is \(x-y\), while the real part of the right side is \(7\). So, we equate them: \(x-y = 7\).
• Imaginary Parts: These are the coefficients of \(i\). On the left, we have \(3i\), and on the right, it's \(yi\). By equating these, we get \(3 = y\).

This step of identifying and equating the corresponding real and imaginary components is crucial in solving equations involving complex numbers.

Solving Equations

Once we have separate equations for the real and imaginary parts, our next task is to solve these equations. Solving equations involves finding values for unknown variables that satisfy all given conditions.
Here's how it works in our example:

Start by tackling imaginary parts first. We've already established that \(3 = y\). This part is straightforward. The solution for \(y\) is simply \(3\). Knowing this helps simplify further processes.

Now, move on to the real parts. You have the equation \(x - y = 7\). Since you already know \(y = 3\), substitute it into the equation:
\[ x - 3 = 7 \]
To solve for \(x\), add \(3\) to both sides:
\[ x = 7 + 3 \]
And there you have it, \(x = 10\). By solving equations in this structured way, you ensure that each variable is accurately determined.

Algebraic Manipulation

Algebraic manipulation is key when working with equations, especially those involving complex numbers. This involves performing operations to isolate variables and simplify the equation, making it easier to solve.

In practice, algebraic manipulation might involve actions like:

• Adding, subtracting, multiplying, or dividing both sides of an equation by the same number.
• Substituting known values to simplify expressions further, which we did by plugging \(y = 3\) into \(x - y = 7\).
• Re-arranging terms to make calculations straightforward.

For instance, in our equation, despite having complex numbers, we split and separately treated their real and imaginary parts under the rules of algebra.
This approach made it much simpler to solve since we zeroed in on specific pieces of the equation at each step.

Algebraic manipulation ensures we derive correct results while maintaining the equation's balance.