= Solution
Row reduction, or the <cofactor> formula for a <matrix inverse>, gives
$$
\boxed{\begin{pmatrix}1&1&1\\1&2&3\\1&4&9\end{pmatrix}^{-1}=\begin{pmatrix}3&-5/2&1/2\\-3&4&-1\\1&-3/2&1/2\end{pmatrix}.}
$$
Multiplying by the right-hand-side column $(5,13,35)^T$ gives, at $\lambda=3$,
$$
\boxed{(x,y,z)=(0,2,3).}
$$
For general $\lambda$, the <coefficient matrix> has <determinant> $(\lambda-1)(\lambda-2)$. Thus other parameter values give a unique solution. If $\lambda=1$, subtracting the first equation from the second gives $y=8$, while subtracting it from the third gives $3y=30$, a contradiction. If $\lambda=2$, subtraction gives $y+z=8$ and $3y+3z=30$, again inconsistent. Hence
$$
\boxed{\text{no solution exactly when }\lambda=1\text{ or }2.}
$$
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