Let $a,b \in \mathbb{Z}$ and suppose that every solution in $\mathbb{Z}$ of the Diophantine equation $$ax+by=35$$ is written as $$x = 7-3t \quad \text{and} \quad y = 7-4t.$$ where $t \in \mathbb{Z}$. Determine the values of $a$ and $b$.
At first, I tried to find the $\gcd(a,b)$ by using the solution of the homogeneous equation, but I got plenty of solutions. Second, if I substitute the equation with the given solutions, I do not know what I can conclude in here. I will appreciate any kind of help. Thank you.