Determine the complex number $z$ satisfying the equation $3z+4i\bar{z}=1-8i$. Note that $\bar{z}$ denotes the conjugate of $z$.

To solve the equation 3z+4izˉ=1−8i, we can use the following steps:

Rewrite the equation in terms of the real and imaginary parts of z. Let z=x+yi, where x is the real part of z and y is the imaginary part of z. Then, the conjugate of z is zˉ=x−yi. Substituting these into the original equation, we get:

3(x + yi) + 4i(x - yi) = 1 - 8i

Expanding the parentheses and multiplying through by i, we get:

(3x + 4y) + (3y - 4x)i = 1 - 8i

Separate the real and imaginary parts of the equation. Equating the real and imaginary parts on both sides of the equation, we get the following system of equations:

3x + 4y = 1 \\ 3y - 4x = -8

Solve the system of equations for x and y. We can solve the system of equations for x and y using any method we like. For example, we can use elimination to solve for x and then back-substitution to solve for y.

Eliminating x from the system of equations, we get:

12y = -7

Dividing both sides by 12, we get:

y = -\frac{7}{12}

Substituting this value of y into the first equation, we get:

3x + 4\left(-\frac{7}{12}\right) = 1

Solving for x, we get:

x = \frac{1}{3}

Therefore, the complex number z satisfying the equation 3z+4izˉ=1−8i is z = 1/3 - 7/12*i.