#function 'c' represents cost
c(x)=60-10*x
#function 'r' represents revenue
r(x)=12*x
#print the value of the cost and revenue with 0 units
print('the cost when the company produces 0 units is')
N(c(0))
print('the revenue when the company produces 0 units is')
N(r(0))
#the value of the cost and revenue with 50 units
print('the cost when the company produces 50 units is')
N(c(50))
print('the revenue when the company produces 50 units is')
N(r(50))
#fuction'p' represents profit
p(x)=c(x)-r(x)
print(p(50))
#graph based on revenue and cost
#cost is shown by graph p1
p1=plot(c(x), -20,20, ymin=-50, ymax=50,gridlines='minor')
#crevenue is shown by graph p2
p2=plot(r(x), -20,20, ymin=-50, ymax=50,gridlines='minor')
show(p1+p2)
#the 

#define new variables
x,y,z=var('x,y,z')
#show the three equations
f(x)=2*x+8*y+4*z==2
k(x)=2*x+5*y+z==5
g(x)=4*x+10*y-z==1
print(f(x))
print(k(x))
print(g(x))
#put the equations into a matrix
B=matrix(3,4,[2,8,4,2,2,5,1,5,4,10,-1,1])
print('the matrix of these three equations is')
print(B)
print('when in reduced echelon form looks like this')
D=B.rref()
print(D)
#there are no rows of zeros which means that every variable has a solution
print('This means the solution to the equations are x=11, y=-4, and z=3')


#create an equation for the circus 
v(x)=(4*x+12*y)/1650==14200
simplify(v(x))