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=2 - Wolfram|Alpha
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=2 - Wolfram|Alpha
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
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: .$\displaystyle \begin{Bmatrix}x &=& 14b + 10 \\ y &=& 3b + 2\end{Bmatrix}\;\text{ for any integer }b.$