Diophantine Equations

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Example

A classic example of a Diophantine equation is $3x + 2y = 11$ .
The goal is to find values for $x$ and $y$ that are whole numbers (integers) and satisfy this equation.

To solve this, we can start by trying different values for $x$ and see if we can find a corresponding value for $y$ that makes the equation true. For example, let's try $x = 1$ :

$3(1) + 2y = 11$
$3 + 2y = 11$
$2y = 11 - 3$
$2y = 8$
$y = \frac{8}{2}$
$y = 4$

So, when $x = 1$ , $y = 4$ . Therefore, one solution to the equation is $x = 1$ and $y = 4$ .

We can check if there are other solutions by trying different values for $x$ . If we try $x = 2$ :

$3(2) + 2y = 11$
$6 + 2y = 11$
$2y = 11 - 6$
$2y = 5$

This doesn't give us an integer solution for $y$ , since 5 divided by 2 is 2.5, not a whole number. Therefore, $x = 2$ doesn't work.

Similarly, if we try $x = 3$ :

$3(3) + 2y = 11$
$9 + 2y = 11$
$2y = 11 - 9$
$2y = 2$
$y = \frac{2}{2}$
$y = 1$

So, another solution is $x = 3$ and $y = 1$ .

Therefore, the Diophantine equation $3x + 2y = 11$ has two positive integer solutions:
$(x, y) = (1, 4)$ and $(x, y) = (3, 1)$ .

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Diophantine Equations

Practise finding integer solutions to equations with more than one unknown.

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