Determine if z is a function of x and y.
1. x^2 z + yz - xy = 10
I say no because there are two z letters in the equation.
2. (x^2/4) + (y^2/9) = 1
I say yes for number 2 but do not know why.
Therefore what?
What does that have to do with determining whether or not z is a function of x and y?
(x^2+ y)z= xy+ 10
z= (xy+ 10)/(x^2+ y).
Now, what does that have to do with determining whether or not z is a function of x and y?
Given values for x and y, how would you calculate a value of z? Did you notice that there is NO z in that formula?2. (x^2/4) + (y^2/9) = 1
I say yes for number 2 but do not know why.
Question 1
I think of a function as something that gives me a unique output value for a given input value. So, in this case, I can think of a function as something that gives me at most one unique value for z in terms of x and y.
I can factor out z and rearrange terms as follows:
z(x^2 + y) - xy = 4
z(x^2 + y) = 4 + xy
z = (4 + xy) / (x^2 + y)
Since I am able to solve for the expression involving x and y that gives me one unique z, I say z is a function of x and y.
Question 2
My goal is to isolate z.
z^2 = 1 - (x^2)/4 + (y^2)/9
I must take the square root on both sides to yield z.
z = sqrt(1 - (x^2)/4 + (y^2)/9)
z = -sqrt(1 - (x^2)/4 + (y^2)/9)
I needed to take the square root on both sides giving TWO answers for z as shown above. I conclude that in this case we do not always get a unique value for z.
So, I say z is not a function of x and y.