# Thread: 3x-14y=2 (solving over integers)

1. ## 3x-14y=2 (solving over integers)

Can somebody please show me how you are supposed to solve this over integers?

3x - 14y = 2

This is what WolframAlpha yields, but it doesn't show any steps.
solve over integers 3x-14y&#61;2 - Wolfram|Alpha

2. ## Re: 3x-14y=2 (solving over integers)

hi

You should be able to do it using the Extended Euclidean Algorithm shown at Extended Euclidean algorithm - Wikipedia, the free encyclopedia

If you have more questions post here.

Stefy

3. ## Re: 3x-14y=2 (solving over integers)

Hello, SweatingBear!

Here is an approaching using only Algebra.

Can somebody please show me how you are supposed to solve this over integers?

. . $\displaystyle 3x - 14y \:=\: 2$

Solve for $\displaystyle x\!:\;\;3x \:=\:14y + 2 \qud\Rightarrow\quad x \:=\:\frac{14y+2}{3}$

So we have: .$\displaystyle x \:=\:4y + \frac{2y+2}{3}$ .[1]

Since $\displaystyle x$ is an integer, $\displaystyle 2y+2$ must be a multiple of 3.
. . $\displaystyle 2y + 2 \:=\:3a\:\text{ for some integer }a.$

Then: .$\displaystyle y \:=\:\frac{3a-2}{2} \quad\Rightarrow\quad y\:=\:a + \frac{a-2}{2}$ .[2]

Since$\displaystyle y$ is an integer, $\displaystyle a-2$ must be a multiple of 2.
. . $\displaystyle a - 2 \:=\:2b \quad\Rightarrow\quad a \:=\:2b+2$

Substitute into [2]: .$\displaystyle y \:=\:(2b+2) + \frac{(2b+2)-2}{2} \quad\Rightarrow\quad y \:=\:3b+2$

Substitute into [1]: .$\displaystyle x \:=\:4(3b+2) + \frac{2(3b+2)+2}{3} \quad\Rightarrow\quad x \:=\:14b + 10$

The solutions are: .$\displaystyle \begin{Bmatrix}x &=& 14b + 10 \\ y &=& 3b + 2\end{Bmatrix}\;\text{ for any integer }b.$