Find integral values of x , y and z : $6x +10y + 15z= -1$

Asked Feb 22 '24 at 15:04

Active Feb 22 '24 at 15:53

Viewed 40 times

I have done similar type of questions before by using the Euclidean Division Lemma/Algorithm to rewrite the equation and then find the solution, but those were problems with only two variables.

In the above equation, as there are three variables, I am confused on how to proceed. Here's what I have tried till now -

$6x + 10y + 15z = -1$

$10y + 3(2x+5z) = -1$

Let $2x+5z = m$. Then, we have $10y + 3m = -1$

For the above equation, we have integral solutions as $y=1$ and $m=-3$.

$2x + 5y = -3$

I am stuck here and unable to find integral solution for x, y. Any help would be greatly appreciated!

edited Feb 22 '24 at 15:53

Bill Dubuque

272,048

asked Feb 22 '24 at 15:04

1025

You already have $y=1$ so $x=-4$... – abiessu Feb 22 '24 at 15:08
if $a,b$ are coprime integers there always exists integers $x,y$ s.t ax+by = 1 (see Bezout's lemma) – Mahammad Yusifov Feb 22 '24 at 15:08
@abiessu How did you get $x= -4$ from $y = 1$? – 1025 Feb 22 '24 at 15:10
$2x+5y=-3$ but the previous solution restricts $y=1$ so then $2x+5=-3$ or $2x=-8$... – abiessu Feb 22 '24 at 15:12
What about $x=1$ and $z=-1$? (It should be $z$, not $y$) – Dominique Feb 22 '24 at 15:12
@MahammadYusifov I have used Bezout's lemma in previous questions, but they only had two variables. How do you use the lemma for 3? – 1025 Feb 22 '24 at 15:12
Oh, yes, the variable mixing is what messed that up. $z=-1, x=1$ works great. – abiessu Feb 22 '24 at 15:13
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@Dominique Thanks! That works! If there is 1 integral solution, there should be a general form too, right? – 1025 Feb 22 '24 at 15:14
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find all $(y,m)$ pairs , then find all $(x,z)$ pairs giving that $m$. – Mahammad Yusifov Feb 22 '24 at 15:16
@MahammadYusifov Thanks! I think I can take it from here. – 1025 Feb 22 '24 at 15:17
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$y=-1+3k,m=3-10k$ (your "$y=1$ and $m=-3$" is wrong) and then $x=-2m+5\ell,z=m-2\ell$. – Anne Bauval Feb 22 '24 at 15:25
@AnneBauval: Shouldn't that be an answer instead of a comment? :-) – Dominique Feb 22 '24 at 15:29
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I am sure @BillDubuque will soon close as a duplicate, with the appropriate links. – Anne Bauval Feb 22 '24 at 15:31

Find integral values of x , y and z : $6x +10y + 15z= -1$

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