Math Is Fun Forum

zetafunc.

Guest

Simultaneous Equations

"Find all solutions in positive integers x, y, z to the simultaneous equations

x + y - z = 12
x² + y² - z² = 12."

Using a substitution, I tried to get an equation involving x and z, which was:

Since x, y and z can only be positive integers, (x - 12) must be a factor of 66. I used this information to get these solutions;

x = 13, y = 78, z = 79
x = 14, y = 43, z = 47
x = 15, y = 34, z = 37
x = 18, y = 33, z = 39
x = 23, y = 18, z = 29
x = 34, y = 15, z = 37
x = 45, y = 14, z = 47
x = 78, y = 13, z = 79

However, I'm not sure if these are the only solutions, and if they are, how can I justify that they are the only solutions? Have I used the wrong approach to this problem?

zetafunc.

Guest

Re: Simultaneous Equations

The second solution should read x = 14, y = 45, z = 47.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Simultaneous Equations

Hi zf

Those look alright. You should have as many solutions as divisors of 66, which is 8.

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bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Simultaneous Equations

Hi zetafunc;

You missed some solutions.

Just joking!

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.