When I type this equation ($6x + 25 = 7y$) into WolframAlpha, it is able to tell me that the integer solution for this equation is:
$x = 7n + 4$, $y = 6n + 7$, where n in the set of all integers
How can I arrive at this solution on my own?
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nonuser
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johnnyodonnell
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3Hint $\ (7 - 6 = 1)\ $ times $25.\ $ Generally use the Extended Euclidean Algorithm to get the Bezout identity $, ja + kb = c,$ for $,\gcd(a,,b) = c,$ then scale that as need be. – Bill Dubuque Jan 29 '19 at 20:34
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Plug in the solution and see what the result is. Can you now do the backwards operation? – Peter Chikov Jan 29 '19 at 20:36
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@PeterChikov what if I don't know the solution? – johnnyodonnell Jan 29 '19 at 20:41
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And see this answer for the form of the general solution (= particular + homogeneous solution) – Bill Dubuque Jan 29 '19 at 20:45
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Thanks @BillDubuque I will take a look at these! – johnnyodonnell Jan 29 '19 at 20:48
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If you plug in the solution you get 42n + 49 = 42n + 49. Working backwards you know that you have to have n's cancel out. So x = 7n + a and y = 6n + b. Plug that in and figure out what a's and b's have to be. – Peter Chikov Jan 29 '19 at 20:56
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This article was helpful: http://mathforum.org/library/drmath/view/51595.html – johnnyodonnell Jan 29 '19 at 22:01
1 Answers
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Since $6\mid 6y-24$ and $6\mid 7y-25$ we have $$6\mid (7y-25)-(6y-24)=y-1$$
Thus $y-1 = 6t$ for some integer $t$, so $\boxed{y= 6t+1}$ and pluging in $6x+25=7y$ we get: $$6x+25 =42t+7\implies \boxed{x= 7t-3}$$
It looks on first sight that I got different solution, but that is not true.
Puting $n=t-1$ we get $$ y=6t+1= 6(t-1)+7=6n+7$$ and $$x= 7(t-1)+4=7n+4$$
nonuser
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What does the vertical pipe symbol mean? Is there an article I can read to learn about that? – johnnyodonnell Jan 30 '19 at 11:07
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$a\mid b$ means $a$ divide $b$, so $3\mid 12$ is true, while $8\mid 4$ is not. – nonuser Jan 30 '19 at 11:08
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The only potential problem I might see with this is that I don't think $7y - 25$ will always be divisible by $6$. For example, when $y = 0$. – johnnyodonnell Feb 01 '19 at 11:09
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