# Can't find 'all' solutions

1. Mar 30, 2005

### recon

I've been asked to find all integer solutions to the following equation.

$$\frac{1}{x} + \frac{2}{y} - \frac{3}{z} = 1$$

Suppose I set y = 2, then it seems to me that there is an infinite number of solutions to the equation.

Is there a systemic way for me to list ALL the integer solutions?

2. Mar 30, 2005

### matt grime

yz+2xz-3xy=xyz

Rearrangements such as

yz=xyz-2xz+3xy

tell you that since x divides the rhs it divides yz, and so on, that may help with any systematic search

3. Mar 30, 2005

### dextercioby

Since u have an equation with 3 unknowns,obviously the # of triplets/sollution is infinite in R.In N,things would go like that

$$x=\frac{yz}{yz-2z+3y}\in \mathbb{N}$$

Daniel.

4. Mar 30, 2005

### Jameson

Is there a way for you to check the answer?

5. Mar 30, 2005

### philosophking

I think you provided a pretty good argument. If you're trying to show that the solutions are infinite, assume one of the variables takes on on value (like you did with y=2), and say that there are an infinite number of x's and z's that solve the remaining equation.

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