= Solution
With $A$ invertible, write the system as $\mathbf u_x+M\mathbf u_y=\mathbf d$, $M=A^{-1}B$. Real distinct <eigenvalues> of $M$ give strict <hyperbolicity>; a real complete <eigenbasis> gives the usual diagonalizable <hyperbolic> form. The <characteristic curves> satisfy $dy/dx=\lambda_i$. Left <eigenvectors> form <characteristic> dependent combinations $w_i=\ell_i\mathbf u$; for constant coefficients they obey separate <transport equations>. <Characteristic> coordinates are functions constant on those curves, such as $y-\lambda_i x$ in the constant case. For variable coefficients extra <derivatives> of the <eigenvectors> can couple the combinations; local <characteristic> directions still follow the real <matrix> pencil. If $A$ is singular one first chooses a noncharacteristic coordinate direction with an invertible coefficient combination.
For the specified constants,
$$
M=\begin{pmatrix}5&3\\-6&-4\end{pmatrix},\qquad\mathbf d=\begin{pmatrix}3\\-2\end{pmatrix}.
$$
Its speeds are $2$ and $-1$, with left <eigenvectors> $(2,1)$ and $(1,1)$. Hence
$$
w_+=2u+v,\quad (w_+)_x+2(w_+)_y=4,\qquad
w_-=u+v,\quad (w_-)_x-(w_-)_y=1.
$$
The transported <Riemann invariants> are $\boxed{R_+=2u+v-4x,\quad R_-=u+v-x}$, constant on $y-2x=\text{constant}$ and $y+x=\text{constant}$, respectively. Initial data give $w_+(0,s)=s-\sin(s/2)$ and $w_-(0,s)=-e^s$, so
$$
w_+=y+2x-\sin(y/2-x),\qquad w_-=-e^{y+x}+x.
$$
Invert these two combinations:
$$
\boxed{u=e^{y+x}+y+x-\sin(y/2-x),\qquad v=-2e^{y+x}-y+\sin(y/2-x).}
$$
Both <characteristic> feet must lie on the supplied positive initial half-line. Thus
$$
\boxed{y-2x>0,\qquad y+x>0,\quad\text{equivalently }y>\max(2x,-x).}
$$
Outside this sector at least one invariant needs unsupplied initial data. The closed expressions give one possible extension there but are not uniquely forced by the question's initial half-line.
\Image[/past-exam-of-the-differential-equations-and-applications-course-of-the-university-of-oxford/2008-hyperbolic-sector.png]
{title=Sector where both backward characteristic families meet the given initial half-line}
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