Problem 37 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 35-38: Find the values of \(x\) and \(y\), where \(x\) and \(y\) are real numbers. $$ (2 x-y)-16 i=10+4 y i $$

Step-by-Step Solution

Answer

\(x = 5\), \(y = -\frac{16}{5}\).

Step 1: Compare Real Parts

We start by comparing the real parts of the complex numbers on each side of the equation. The real part on the left is \(2x\) and on the right is \(10\). Therefore, the equation for the real part is \(2x = 10\).

Step 2: Solve for \(x\)

From the equation \(2x = 10\), we can solve for \(x\) by dividing both sides by 2: \(x = \frac{10}{2} = 5\).

Step 3: Compare Imaginary Parts

Next, we compare the imaginary parts of the complex numbers. The imaginary part on the left is \(-y - 16\) and on the right is \(4y\). Therefore, the equation for the imaginary part is \(-y - 16 = 4y\).

Step 4: Solve for \(y\)

From the equation \(-y - 16 = 4y\), we first add \(y\) to both sides to get \(-16 = 5y\). Then, we divide both sides by 5: \(y = \frac{-16}{5}\).

Key Concepts

Real and Imaginary Parts

When dealing with complex equations, it's essential to separate the real and imaginary parts. Complex numbers can be expressed in the form of \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part. In our exercise, we need to find the values of \(x\) and \(y\), which are both real numbers.
We start by examining the given equation, \((2x - y) - 16i = 10 + 4yi\). Separate the equation into its real and imaginary components:

• The real part on the left side is \(2x \)
• On the right side, the real part is \(10\)

Therefore, we obtain \(2x = 10\). For the imaginary parts, we observe:

• The imaginary part on the left is \(-y - 16\)
• On the right, it is \(4y\)

This gives us the equation \(-y - 16 = 4y\). By understanding these components, solving becomes straightforward.

System of Equations

Once we separate the real and imaginary components, we create a system of equations from them. A system of equations involves finding values that satisfy more than one equation simultaneously. Here, we have two equations for our unknowns, \(x\) and \(y\):

• The real equation: \(2x = 10\)
• The imaginary equation: \(-y - 16 = 4y\)

Solving a system of equations requires addressing each equation step-by-step to find values for the variables.
For the real part, solving \(2x = 10\) is straightforward by dividing by 2. Whereas solving \(-y - 16 = 4y\) involves algebraic manipulation that we'll explore next. Each equation gives us a vital clue about one of the unknowns.

Algebraic Manipulation

To find the values of \(x\) and \(y\), we need to manipulate the equations algebraically. For \(x\), the equation \(2x = 10\) simplifies directly to \(x = 5\) by dividing both sides by 2.
For the imaginary part equation \(-y - 16 = 4y\), start by adding \(y\) to both sides:

• This results in \(-16 = 5y\).

Now, solve for \(y\) by dividing both sides by 5, which gives \(y = \frac{-16}{5}\).
These algebraic manipulations are crucial. They allow us to isolate each variable and solve the system of equations step-by-step, ensuring that both real and imaginary parts are balanced.