The line with equation $y = 3x + 5$ has slope 3 and $y$-intercept 5. Since the line has $y$-intercept 5, it passes through $(0, 5)$. When the line is translated 2 units to the right, its slope does not change and the new line passes through $(2, 5)$. A line with slope $m$ that passes through the point $(x_1, y_1)$ has equation $y - y_1 = m(x - x_1)$. Therefore, the line with slope 3 that passes through $(2, 5)$ has equation $y - 5 = 3(x - 2)$ or $y - 5 = 3x - 6$, which gives $y = 3x - 1$. Alternatively, we could note that when the graph of $y = 3x + 5$ is translated 2 units to the right, the equation of the new graph is $y = 3(x - 2) + 5$ or $y = 3x - 1$.