Find all solutions to the equation $$101x + 97y = 2.$$ Supposed answer is $x = -24 +97k$, $y = 25-101k$. I am unsure what steps to take to find these solutions.
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The correct form of your equation is $101x +97y=1$ if the solutions are correct. You must use homogeneous equation for general solutions; if equation is $ax+by=c$ then it's homogeneous equation is $ax+by=0$. the solutions of this equation are: $x=b$ and $y=-a$, so the general form of solutions is:
Where $x_1$ and $y_1$ are one solution of initial equation $101 x +97y=1$. you can see that $x_1=-24$ and $y_1=25$ are one solution, so general solutions are:
$x=97 k -24$
$y=-101 k +25$
Finding primitive solutions: $101x +97y=1$ indicates that $101 x$ and $97 y$ are two consecutive numbers with opposite signs; One odd and it's absolute value greater than the other which is even. Suppose:
$97 y-101 x=1$
$97 y +101 x= 4k+1$
We rewrite this relation as:
$(4\times 24+1) y +(4\times 25+1)x=4(24 y+ 25 x) +x+y=4k+1$
We may assume:
$24y+25x=k$; which homogeneous equation is $24y+25x=0$ and gives $y=25, x=-24$ and $x+y=1$ which is also satisfied with these values of x and y. So $x=-24, y=25$ can be accepted as primitive solutions.