https://file.aiquickdraw.com/mathbot/file/1728936132885vii2ueou.jpg

To simplify the system of equations from the image, let's first break down both equations and simplify step by step.

### Given equations:
1. \[
\frac{20 - V_1}{10} = \frac{V_1}{-j2.5} + \frac{V_1 - V_2}{j4}
\]
2. \[
(1 + j1.5)V_1 + j2.5V_2 = 20
\]

### Steps for Simplification of Equation 1:
\[
\frac{20 - V_1}{10} = \frac{V_1}{-j2.5} + \frac{V_1 - V_2}{j4}
\]

1. Multiply through by 10 to eliminate the denominator on the left-hand side:
   \[
20 - V_1 = \frac{10V_1}{-j2.5} + \frac{10(V_1 - V_2)}{j4}
   \]

2. Simplify the terms involving \( j \):
   - \[
   \frac{10V_1}{-j2.5} = -\frac{4V_1}{j} = 4jV_1
   \]
   - \[
   \frac{10(V_1 - V_2)}{j4} = \frac{2.5(V_1 - V_2)}{j} = -2.5j(V_1 - V_2)
   \]

So, the equation becomes:
\[
20 - V_1 = 4jV_1 - 2.5j(V_1 - V_2)
\]

### Simplification of Equation 2:
\[
(1 + j1.5)V_1 + j2.5V_2 = 20
\]

This equation is already in its simplest form, but we can expand it if needed.

Would you like me to proceed with solving the system of equations, or provide further simplification steps?

### Follow-up Questions:
1. Would you like help solving for \(V_1\) and \(V_2\) from these simplified equations?
2. Should we focus on solving the real and imaginary components separately?
3. Would you prefer a detailed explanation of how complex numbers affect the simplification process?
4. Do you need a breakdown of how I simplified the terms involving \(j\)?
5. Should I rearrange the second equation for easier substitution?

### Tip:
When dealing with complex numbers in equations, it's often helpful to separate real and imaginary parts early to simplify the solving process.

Learn how to simplify complex algebraic equations involving imaginary numbers. This problem involves breaking down and simplifying the given equations step-by-step. Suitable for students in Grades 11-12.

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