differentiation problem help/check work

find all values of the constants a and b for which the function

f(x)=ax, x<2 and ax^2-bx+3, x greater than or equal to 2

will be differentiable for all values of x

so what i first did was to find the values of a and b for which the function is continuous, so i took the left sided and right sided limits as they approached 2

for the left sided limit i got 2a, and for the right sided limit i got 4a-2b+3

i set them equal to each other 2a=4a-2b+3 and rearranged to get -2a+2b=3

next i found the left sided and right sided limits of the derivative of the function as it approached 2

for the left side limit i got 2, and for the right side limit i got 4a-b

setting them equal to each other to get 4a-b=2, i then have a system of equations to solve

-2a+2b=3

4a-b=2

solving i get a=7/6 and b=8/3

i just want someone to look over my work and verify that its right or wrong, or if this is how someone would go about solving a problem like this, i wasnt really sure, my first guess was that i had to use the limit definition of the derivative to somehow get the answer, but i wasnt really sure if that would work so i tried the method used above.