$$ \begin{aligned} \frac{4}{z}+\frac{1}{z} &= \frac{2}{z}+3&& \text{multiply ALL terms by } \color{blue}{ z }. \\[1 em]z\cdot\frac{4}{z}+z\cdot\frac{1}{z} &= z\cdot\frac{2}{z}+z\cdot3&& \text{cancel out the denominators} \\[1 em]4+1 &= 2+3z&& \text{simplify left and right hand side} \\[1 em]5 &= 3z+2&& \text{ swap the left and right side of the equation } \\[1 em]3z+2 &= 5&& \text{ move 2 to the right } \\[1 em]3z &= 5-2&& \\[1 em]3z &= 3&& \text{ divide both sides by $ 3 $ } \\[1 em]z &= \frac{3}{3}&& \\[1 em]z &= 1&& \\[1 em] \end{aligned} $$

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