CEMC Cayley, 2024

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
$y = 3x-1$.