Let x and y represent the prices of one plate and one cup.
Equation 1:
4x+3y=371
Equation 2:
x-y=14
Using equation 2, solve for y in terms of x.
y=x-14
By substitution, solve for x using equation 1.
4x+3\left(x-14\right)=371,
7x=413,
x=\frac{413}{7}=59
So, each plate costs $59.