# Solving Two-Variable Problems: F(x,y)=-(y/x)^2+h(xy)

• Bleys
In summary, there are some problems that ask you to find an equation with given information. In this specific conversation, the problem is to find a function h that satisfies the given conditions. After some discussion and substitution, it is discovered that h(y) = 0 is a solution to the problem. However, it is noted that there could be multiple ways to express h and it is unclear what the explicit form of h should be. The conversation ends with a question about the meaning of h(y)=0 in relation to the restricted domain of (1,y). Overall, the conversation highlights the process of solving this type of problem and the potential for multiple solutions.
Bleys
There are some problems that ask you to find an equation with some information and I'm having trouble with it
For example, z= F(x,y) = -(y/x)^2 + h(xy), for some arbitrary function h.
They give you the condition that F(1,y) = y^2, and to find z now.
So with obvious substitution you get h(y) = 0
But how does that tell you anything else about z?
Does it mean h(t) = (t - y) ? Aren't there an infinite amount of ways to express h anyway (like ln(t-y+1) for example)? I doubt they are asking you for an explicit form, since no other information is available, but how do you go about solving a problem like this. I'm having a little problems understanding what exactly it's saying.
Does h(y)=0 mean y is a solution to the function h when the domain is restricted to points (1,y)?

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Looking carefully at your function when you insert x=1 you have $$F(1,y)=-y^2 + H(1,y)$$ if the negative sign is in the right place. The answer should be obvious what $$H(x,y)$$ is

Bleys said:
There are some problems that ask you to find an equation with some information and I'm having trouble with it
For example, z= F(x,y) = -(y/x)^2 + h(xy), for some arbitrary function h.
They give you the condition that F(1,y) = y^2, and to find z now.
So with obvious substitution you get h(y) = 0
But how does that tell you anything else about z?
Does it mean h(t) = (t - y) ? Aren't there an infinite amount of ways to express h anyway (like ln(t-y+1) for example)? I doubt they are asking you for an explicit form, since no other information is available, but how do you go about solving a problem like this. I'm having a little problems understanding what exactly it's saying.
Does h(y)=0 mean y is a solution to the function h when the domain is restricted to points (1,y)?
h is a function of a single variable- its domain is a subset of the set of all real numbers, not points in the plane.

F(1,y)= -(y/1)^2+ h(1y)= -y^2+ h(y)= y^2, but that does NOT give you "h(y)= 0" it gives "h(y)= 2y^2". F(x,y)= -(y/x)^2+ 2(xy)^2 for all x and y.

Did you mean F(x,y)= (y/x)^2+ h(xy) and F(1,y)= y^2? In that case F(1,y)= (y/1)^2+ h(1y)= y^2 so h(y)= 0 for all y and so F(x,y)= (y/x)^2 for all x and y.

Did you mean F(x,y)= -(y/x)^2+ h(xy) and F(1,y)= -y^2? In that case F(1,y)= -(y/1)+ h(1y)= -y^2 so again h(y)= 0. F(x,y)= -(y/x) for all x and y.

## 1. What is a two-variable problem?

A two-variable problem is a mathematical problem that involves two different variables, usually represented by x and y. In these types of problems, the value of one variable is dependent on the value of the other variable.

## 2. How do I solve a two-variable problem?

To solve a two-variable problem, you will need to use algebraic techniques such as substitution or elimination. You will also need to have equations for both variables in terms of each other, which is often given in the problem.

## 3. What does F(x,y) represent in this equation?

F(x,y) represents a function that has two variables, x and y. This means that the function's output, or value, will change depending on the values of x and y that are inputted into the function.

## 4. What does h represent in this equation?

In this equation, h represents a constant or coefficient that is multiplied by the product of xy. This can potentially affect the shape and location of the graph of the function.

## 5. What is the significance of -(y/x)^2 in this equation?

The term -(y/x)^2 represents the inverse relationship between x and y in this equation. This means that as one variable increases, the other variable will decrease in proportion. This can also affect the curvature and direction of the graph of the function.

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