Question

Solve for all possible values of the real numbers \(x\) and \(y\) in the following equations. $$(2 x-3 y-5)+i(x+2 y+1)=0$$

Short answer

The possible values of \(x\) and \(y\) are \(x = 1\) and \(y = -1\).

Step-by-step solution

- Equate real parts

Separate the real part from the equation \((2x - 3y - 5) = 0\) and set it equal to zero.

- Solve real part equation

Solve the equation obtained from the real part: \(2x - 3y - 5 = 0\). Rearrange it to find \(x\) in terms of \(y\). \[ x = \frac{3y + 5}{2} \]

- Equate imaginary parts

Separate the imaginary part from the equation \((x + 2y + 1)i = 0\) and set it equal to zero (since the imaginary part must also be zero).

- Solve imaginary part equation

Solve the equation obtained from the imaginary part: \(x + 2y + 1 = 0\). Substitute the value of \(x\) found earlier into this equation. \[ \left( \frac{3y + 5}{2} \right) + 2y + 1 = 0 \]

- Simplify and solve for y

Combine the fractions and solve for \(y\). \[ \frac{3y + 5 + 4y + 2}{2} = 0 \] \[ 7y + 7 = 0 \] \[ 7y = -7 \] \[ y = -1 \]

- Substitute y and find x

Substitute \(y = -1\) back into the equation for \(x\) obtained in Step 2. \[ x = \frac{3(-1) + 5}{2} = \frac{-3 + 5}{2} = \frac{2}{2} = 1 \]

Key concepts

Real Part

In this exercise, the term 'real part' refers to the part of a complex expression that does not contain the imaginary unit \(i\).

Given a complex number, such as \((2x - 3y - 5) + i(x + 2y + 1)\), the real part is the expression without \(i\). Here, it's \(2x - 3y - 5\).

To solve for the real part, set it equal to zero:
\(2x - 3y - 5 = 0\).
This step lets us isolate one of the variables—\(x\) or \(y\) in this case.

Problem-solving often requires isolating variables to simplify complex equations. For example, from \(2x - 3y - 5 = 0\), you can rearrange the terms to express \(x\) in terms of \(y\): \[ x = \frac{3y + 5}{2} \]
This equation is crucial for solving the next part of the problem.

Imaginary Part

The imaginary part of a complex number includes the imaginary unit \(i\). In our problem, the imaginary part is \(x + 2y + 1\).
Just like with the real part, you want this part to be zero, ensuring the entire equation is equal to zero.

So, we set it up as: \(x + 2y + 1 = 0\).
Here, we substitute \(x\) using the expression we found earlier: \[ \left( \frac{3y + 5}{2} \right) + 2y + 1 = 0 \]
Simplifying and combining like terms will help you solve for \(y\).

Combining fractions effectively lets you solve complex equations step-by-step.

Solving Equations

To solve the given equations, you need to bring together the solutions for both real and imaginary parts.
After combining fractions, the next step is simplifying the equation:
\[ \frac{3y + 5 + 4y + 2}{2} = 0 \] leads to \[ 7y + 7 = 0 \]
Then solve for \(y\): \[ y = -1 \]

Substituting \(y\) back into the equation for \(x\): \[ x = \frac{3(-1) + 5}{2} = 1 \]
The final values are: \( x = 1 \) and \( y = -1 \).

This clear, step-by-step method will help in solving many such equations. Understanding these basics can help tackle more complex problems in your learning journey.