Answer
$P(2 / 5,19 / 5)$ is the closest point to $P(4,2)$ that is on the given line $2 x+3=y$
Work Step by Step
From $\nabla f=\lambda \nabla g$
\[
\begin{aligned}
&\langle 2(x-4), 2(y-2)\rangle=\lambda\langle-2,1\rangle \\
\Rightarrow \lambda=& \frac{2(x-4)}{-2}=2(y-2) \equiv 4-x=2(y-2) \Rightarrow x=8-2 y
\end{aligned}
\]
Using this in the given constraint
\[
y=2 x+3 \equiv y=2(8-2 y)+3 \Rightarrow 5 y=19 \Rightarrow y=19 / 5
\]
Then
\[
8-2 y=\frac{40-2 \cdot 19}{5}=2 / 5=x
\]
Hence, $P(2 / 5,19 / 5)$ is the closest point to $P(4,2)$ that is on the given line $2 x+3=y$
