$$\frac{3z+2}{3}=\frac{5z-2}{4}$$
$z = \frac{14}{3} = 4\frac{2}{3} \approx 4.666666667$
$$4\left(3z+2\right)=3\left(5z-2\right)$$
$$12z+8=3\left(5z-2\right)$$
$$12z+8=15z-6$$
$$12z+8-15z=-6$$
$$-3z+8=-6$$
$$-3z=-6-8$$
$$-3z=-14$$
$$z=\frac{-14}{-3}$$
$$z=\frac{14}{3}$$
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