The roots of the quadratic equation $x^2 - 5x + 3 = 0$ are $\alpha$ and $\beta$. Without solving the equation, find the exact value of $\frac{1}{\alpha} + \frac{1}{\beta}$.

1. 利用根与系数的关系写出两根之和与两根之积

对于二次方程 $x^2 - 5x + 3 = 0$，由根与系数的关系可得：
$\alpha + \beta = -\frac{-5}{1} = 5$
$\alpha \beta = \frac{3}{1} = 3$

2. 将待求代数式通分并代入数值计算

将 $\frac{1}{\alpha} + \frac{1}{\beta}$ 通分改写为以 $\alpha+\beta$ 与 $\alpha\beta$ 表示的形式：
$\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta}$
代入 $\alpha + \beta = 5$ 和 $\alpha \beta = 3$：
$\frac{1}{\alpha} + \frac{1}{\beta} = \frac{5}{3}$