The problem given in the image is written in Bengali and involves solving for the values of $x$ and $y$ in a complex equation. Here's a translation and breakdown of the mathematical content:

The equation presented is: $\frac{(1+i)x - 2i}{3+i} + \frac{(2-3i)y + i}{3-i} = i$

To solve this, we will follow these steps:

Step 1: Simplify the Complex Denominators

First, rationalize the denominators by multiplying both the numerator and denominator of each fraction by the conjugate of the denominator.

For the first term: $\frac{(1+i)x - 2i}{3+i} \cdot \frac{3-i}{3-i} = \frac{[(1+i)x - 2i](3-i)}{(3+i)(3-i)}$ Since $(3+i)(3-i) = 9 - i^2 = 9 + 1 = 10$, the denominator becomes 10.

For the second term: $\frac{(2-3i)y + i}{3-i} \cdot \frac{3+i}{3+i} = \frac{[(2-3i)y + i](3+i)}{(3-i)(3+i)}$ Similarly, $(3-i)(3+i) = 9 + 1 = 10$, so the denominator is again 10.

Step 2: Expand the Numerators

Now, expand the numerators for both terms.

For the first term: $[(1+i)x - 2i](3-i) = (1+i)x(3-i) - 2i(3-i)$ Expanding each term: $(1+i)x(3-i) = (3 - i + 3i + 1)x = (4 + 2i)x$ $-2i(3-i) = -6i + 2 = 2 - 6i$ Thus, the first term's numerator becomes: $(4+2i)x + (2-6i)$

For the second term: $[(2-3i)y + i](3+i) = (2-3i)y(3+i) + i(3+i)$ Expanding each term: $(2-3i)y(3+i) = (6 - 9i + 2i - 3)y = (6-7i)y$ $i(3+i) = 3i - 1$ Thus, the second term's numerator becomes: $(6-7i)y + (3i - 1)$

Step 3: Combine the Equation

Now combine both terms over 10: $\frac{(4+2i)x + (2-6i)}{10} + \frac{(6-7i)y + (3i-1)}{10} = i$ Multiply through by 10 to eliminate the denominators: $(4+2i)x + (2-6i) + (6-7i)y + (3i-1) = 10i$

Step 4: Collect Real and Imaginary Parts

Group the real and imaginary parts separately:

• Real part: $4x + 2 + 6y - 1 = 0$, which simplifies to: $4x + 6y + 1 = 0$

• Imaginary part: $2ix - 6i - 7iy + 3i = 10i$, which simplifies to: $2x - 6 - 7y + 3 = 10$ Simplifying further: $2x - 7y - 3 = 10$ $2x - 7y = 13$

Step 5: Solve the System of Equations

Now, solve the system of equations:

1. $4x + 6y = -1$
2. $2x - 7y = 13$

You can use substitution or elimination to solve this system for $x$ and $y$.

Would you like me to proceed with solving the system of equations, or do you have any questions about this process so far?

Relative Questions:

1. What is the next step after collecting real and imaginary parts in complex equations?
2. How can we use the conjugate method to simplify complex fractions?
3. How do you interpret and solve systems of linear equations?
4. Why is it important to separate real and imaginary parts in complex equations?
5. Can this method be generalized to more complex equations with multiple variables?

Tip:

Always double-check your expansions when dealing with complex numbers to ensure both real and imaginary parts are correctly handled.