Answer: Fran's number is 70
Step-by-step explanation:
Let x represent Jen's number.
Let y represent Carrie's number.
Let z represent Fran's number.
When you add their numbers together you get 207. This means that
x + y + z = 207 - - - - - - - - - -1
Jen's number is 9 more than Carrie's number. It means that
y = x - 9
Fran's number is 3 less than Jen's number. It means that
z = x - 3
Substituting y = x - 9 and z = x - 3 into equation 1, it becomes.
x + x - 9 + x - 3 = 207
3x - 12 = 207
3x = 207 + 12 = 219
x = 219/3 = 73
y = x - 9 = 73 - 9
y = 64
z = x - 3 = 73 - 3
z = 70
I don’t think this exact answer is there
We have
![(-3xy^2+y)_y=--6xy+1](https://tex.z-dn.net/?f=%28-3xy%5E2%2By%29_y%3D--6xy%2B1)
and
![(-3x^2y+x)_x=-6xy+1](https://tex.z-dn.net/?f=%28-3x%5E2y%2Bx%29_x%3D-6xy%2B1)
so the equation is indeed exact. So we want to find a function
such that
![F_x=-3xy^2+y](https://tex.z-dn.net/?f=F_x%3D-3xy%5E2%2By)
![F_y=-3x^2y+x](https://tex.z-dn.net/?f=F_y%3D-3x%5E2y%2Bx)
Integrating both sides of the first equation wrt
gives
![F(x,y)=-\dfrac32x^2y^2+xy+f(y)](https://tex.z-dn.net/?f=F%28x%2Cy%29%3D-%5Cdfrac32x%5E2y%5E2%2Bxy%2Bf%28y%29)
Differentiating both sides wrt
gives
![F_y=-3x^2y+x=-3x^2y+x+f_y\implies f_y=0\implies f(y)=C](https://tex.z-dn.net/?f=F_y%3D-3x%5E2y%2Bx%3D-3x%5E2y%2Bx%2Bf_y%5Cimplies%20f_y%3D0%5Cimplies%20f%28y%29%3DC)
So we have
![F(x,y)=-\dfrac32x^2y^2+xy+C=C](https://tex.z-dn.net/?f=F%28x%2Cy%29%3D-%5Cdfrac32x%5E2y%5E2%2Bxy%2BC%3DC)
or
![F(x,y)=\boxed{-\dfrac32x^2y^2+xy=C}](https://tex.z-dn.net/?f=F%28x%2Cy%29%3D%5Cboxed%7B-%5Cdfrac32x%5E2y%5E2%2Bxy%3DC%7D)