Understanding the Concept:
To find the conjugate of the multiplicative inverse of a complex number:
\[
z^{-1}=\frac1z
\]
and then take the complex conjugate.
The conjugate of:
\[
a+bi
\]
is:
\[
a-bi
\]
Step 1: Simplify the given complex number.
Given:
\[
z=\frac{1+7i}{3+i}
\]
Multiply numerator and denominator by the conjugate of the denominator:
\[
3-i
\]
Thus,
\[
z=
\frac{(1+7i)(3-i)}{(3+i)(3-i)}
\]
Step 2: Expand numerator and denominator.
Numerator:
\[
(1+7i)(3-i)
\]
\[
=3-i+21i-7i^2
\]
Since
\[
i^2=-1,
\]
\[
=3+20i+7
\]
\[
=10+20i
\]
Denominator:
\[
(3+i)(3-i)=9+1=10
\]
Therefore,
\[
z=\frac{10+20i}{10}
\]
\[
=1+2i
\]
Step 3: Find the multiplicative inverse.
\[
z^{-1}=\frac1{1+2i}
\]
Multiply numerator and denominator by the conjugate:
\[
1-2i
\]
\[
z^{-1}
=
\frac{1-2i}{1+4}
\]
\[
=\frac15-\frac25i
\]
Step 4: Take the conjugate.
The conjugate is:
\[
\frac15+\frac25i
\]
Hence,
\[
\boxed{\frac15+\frac25i}
\]