Non-linear system of equations involving sine

Question

I would like to know if it is possible to solve for $x$ and $y$ in $$\begin{cases} a\sin(bx+y)=c\\ dx+y=f \end{cases} $$ in terms of real constants $a,b,c,d,f$.

Answer 1

Hint:

Let $t:=bx+y$ and rewrite the system as

$$\begin{cases}\dfrac bd(t-y)+y=f,\\\sin(t)=\dfrac ca\end{cases}.$$

Solve for $t$, then deduce $y$ and $x$.