The gas phase reaction X+Y⇄ Z is first order in Y and second order in X. If the concentration of X is double and concentration of Y is tripled. The reaction rate would be increase by a factor of:

then,
Rate= k [x]^2[y] --> k[x]^4[y}^3

is this the correct line of thought? from this rate law how would one determine the amount that the rate is increased?

2 answers

The right line of thought but you missed a turn somewhere.
That's rate = k[x]^2[y]^1, then
rate = k(2)^2(3)^1 = 12*k
rate=k [x]^2[y]

if you double x, then rate factor goes up by 4, and triple y, up by 3

overall, the rate has increased by 12

No your line of thought is nor correct.