Let a and c represent, respectively, the price of an adult and a child ticket. We know that
Multiply the first equation by 3 and the second equation by 2 to get
Subtract the two equations to have
Plug this value in the first equation to get
The local weather report states that there is more than a 2/3 chance of rain for Saturday.
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We want to find a scalar function 
such that 
.
So we need to have
Integrating both sides with respect to
gives
Differentiating with respect to
gives
So we find that
By the fundamental theorem of calculus, we then know the line integral depends only on the values of at the endpoints of the path. Therefore
So we need to have
Integrating both sides with respect to
Differentiating with respect to
So we find that
By the fundamental theorem of calculus, we then know the line integral depends only on the values of