= Solution
Write $d=\mu-r\mathbf1$ and let $x$ denote the risky holdings as fractions of total wealth. The remaining fraction $1-\mathbf1^Tx$ is in the risk-free asset. Consequently their excess <expected return> is $d^Tx$ and their <variance> is $x^T\Sigma x$. The <Cauchy-Schwarz inequality> in the <inner product> defined by $\Sigma$ gives
$$
d^Tx=(\Sigma^{-1}d)^T\Sigma x
\leq\sqrt{d^T\Sigma^{-1}d}\sqrt{x^T\Sigma x}.
$$
Equality with positive excess return holds precisely for $x=cz$, $c>0$, where $z=\Sigma^{-1}d$. Thus solving $\Sigma z=d$ determines both the efficient risky direction and the upper <capital market line> slope. Provided $s=\mathbf1^Tz>0$, the normalized <market portfolio> and its <Sharpe ratio> are
$$
\boxed{w_M=\frac{z}{s},\qquad \theta=\sqrt{d^Tz}.}
$$
Here the excess means are $(7,14,21)^T$, so the required equations are
$$
4z_1+z_2+z_3=7,\quad z_1+9z_2+2z_3=14,\quad z_1+2z_2+16z_3=21.
$$
The leading principal minors of $\Sigma$ are $4$, $35$ and $539$, so <Sylvester's criterion> verifies positive definiteness and the solution is unique.
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