First, we subtract 19 from both sides to get $x^2-4x-12=0$.
Then, we factor the quadratic as $(x-6)(x+2)=0$.
So, the solutions are $x=6$ and $x=-2$, with $a=6$ and $b=-2$.
Finally, we calculate $2a+b = 2(6)+(-2) = 12-2 = \boxed{10}$.
The answer is: 10
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
The equation $x^2-4x+7=19$ has two solutions, $a$ and $b$, with $a\geq b$. What is the value of $2a+b$?

### Answer:
