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If $\dfrac{z-12}{z-8i}=\dfrac{5}{3},$ set the value of the complex number $z.$

I started to multiply the complex fraction by the conjugate which I believe is $\dfrac{\bar z + 8i}{\bar z + 8i}.$

My problem is that the terms of the multiplication are getting hard, because I got thing like $|z|$.
Then I guess that the real part is going to be $\frac{5}{3}$ and the imaginary one zero.

Thanks.

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    $\begingroup$ Hint: $3(z-12) = 5(z-8i)$ $\endgroup$ – Hyperion May 31 at 0:22
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You're overthinking a bit. Reduce this to standard algebra; solve for $z$. Recall your exercises with similar functions from middle school in terms of a real variable $x$.

I'll work out a similar problem to avoid giving you a direct answer to the given problem.

Find $z$ if $$\frac{z-2i}{z-3i} = \frac{5}{7}$$

We begin by "cross multiplying" - multiply both sides by the denominators of each side. Thus we get

$$\frac{z-2i}{z-3i} = \frac{5}{7} \implies 7(z-2i)=5(z-3i) \implies 7z-14i =5z-15i$$

Now we solve for $z$ by subtracting $5z$ from both sides, and adding $14i$ to both sides:

$$7z-14i =5z-15i \implies 2z = -i \implies z = - \frac 1 2 i$$

As a result, we get that

$$z = - \frac 1 2 i = 0 - \frac 1 2 i$$

whichever form you desire to use. A similar idea holds for your given problem

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