How to solve the equation $15x- 16y= 10$ [duplicate]
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This question is an exact duplicate of:
Find the value of $x,y$ for the equation $12x+5y=7$ using number theory .
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I am trying to find an $x$ and $y$ that solve the equation $15x - 16y = 10$, usually in this type of question I would use Euclidean Algorithm to find an $x$ and $y$ but it doesn't seem to work for this one. Computing the GCD just gives me $16 = 15 + 1$ and then $1 = 16 - 15$ which doesn't really help me. I can do this question with trial and error but was wondering if there was a method to it.
Thank you
algebra-precalculus euclidean-algorithm
edited Nov 16 at 22:18
Lord Shark the Unknown
98.2k958131
asked Nov 16 at 22:15
ElMathMan
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marked as duplicate by Bill Dubuque algebra-precalculus
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Nov 16 at 22:51
This question was marked as an exact duplicate of an existing question.
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This question is an exact duplicate of:
Find the value of $x,y$ for the equation $12x+5y=7$ using number theory .
2 answers
I am trying to find an $x$ and $y$ that solve the equation $15x - 16y = 10$, usually in this type of question I would use Euclidean Algorithm to find an $x$ and $y$ but it doesn't seem to work for this one. Computing the GCD just gives me $16 = 15 + 1$ and then $1 = 16 - 15$ which doesn't really help me. I can do this question with trial and error but was wondering if there was a method to it.
Thank you
algebra-precalculus euclidean-algorithm
edited Nov 16 at 22:18
Lord Shark the Unknown
98.2k958131
asked Nov 16 at 22:15
ElMathMan
193
marked as duplicate by Bill Dubuque algebra-precalculus
Users with the algebra-precalculus badge can single-handedly close algebra-precalculus questions as duplicates and reopen them as needed.
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Nov 16 at 22:51
This question was marked as an exact duplicate of an existing question.
Can you solve the congruence $15xequiv10pmod{16}$?
– Lord Shark the Unknown
Nov 16 at 22:17
$x$ is congruent to $6$ (mod $16$) ? but not sure how that helps me? thanks
– ElMathMan
Nov 16 at 22:30
$xequiv6pmod{16}$ means $x=6+16k$ where $kinBbb Z$. So then, what is $y$?
– Lord Shark the Unknown
Nov 17 at 5:20
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
This question is an exact duplicate of:
Find the value of $x,y$ for the equation $12x+5y=7$ using number theory .
2 answers
I am trying to find an $x$ and $y$ that solve the equation $15x - 16y = 10$, usually in this type of question I would use Euclidean Algorithm to find an $x$ and $y$ but it doesn't seem to work for this one. Computing the GCD just gives me $16 = 15 + 1$ and then $1 = 16 - 15$ which doesn't really help me. I can do this question with trial and error but was wondering if there was a method to it.
Thank you
algebra-precalculus euclidean-algorithm
edited Nov 16 at 22:18
Lord Shark the Unknown
98.2k958131
asked Nov 16 at 22:15
ElMathMan
193
This question is an exact duplicate of:
Find the value of $x,y$ for the equation $12x+5y=7$ using number theory .
2 answers
I am trying to find an $x$ and $y$ that solve the equation $15x - 16y = 10$, usually in this type of question I would use Euclidean Algorithm to find an $x$ and $y$ but it doesn't seem to work for this one. Computing the GCD just gives me $16 = 15 + 1$ and then $1 = 16 - 15$ which doesn't really help me. I can do this question with trial and error but was wondering if there was a method to it.
Thank you
This question is an exact duplicate of:
Find the value of $x,y$ for the equation $12x+5y=7$ using number theory .
2 answers
algebra-precalculus euclidean-algorithm
algebra-precalculus euclidean-algorithm
edited Nov 16 at 22:18
Lord Shark the Unknown
98.2k958131
asked Nov 16 at 22:15
ElMathMan
193
edited Nov 16 at 22:18
Lord Shark the Unknown
98.2k958131
asked Nov 16 at 22:15
ElMathMan
193
edited Nov 16 at 22:18
Lord Shark the Unknown
98.2k958131
edited Nov 16 at 22:18
Lord Shark the Unknown
98.2k958131
edited Nov 16 at 22:18
Lord Shark the Unknown
98.2k958131
98.2k958131
asked Nov 16 at 22:15
ElMathMan
193
asked Nov 16 at 22:15
ElMathMan
193
asked Nov 16 at 22:15
ElMathMan
193
193
marked as duplicate by Bill Dubuque algebra-precalculus
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Nov 16 at 22:51
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Nov 16 at 22:51
This question was marked as an exact duplicate of an existing question.
Can you solve the congruence $15xequiv10pmod{16}$?
– Lord Shark the Unknown
Nov 16 at 22:17
$x$ is congruent to $6$ (mod $16$) ? but not sure how that helps me? thanks
– ElMathMan
Nov 16 at 22:30
$xequiv6pmod{16}$ means $x=6+16k$ where $kinBbb Z$. So then, what is $y$?
– Lord Shark the Unknown
Nov 17 at 5:20
add a comment |
Can you solve the congruence $15xequiv10pmod{16}$?
– Lord Shark the Unknown
Nov 16 at 22:17
$x$ is congruent to $6$ (mod $16$) ? but not sure how that helps me? thanks
– ElMathMan
Nov 16 at 22:30
$xequiv6pmod{16}$ means $x=6+16k$ where $kinBbb Z$. So then, what is $y$?
– Lord Shark the Unknown
Nov 17 at 5:20
Can you solve the congruence $15xequiv10pmod{16}$?
– Lord Shark the Unknown
Nov 16 at 22:17
Can you solve the congruence $15xequiv10pmod{16}$?
– Lord Shark the Unknown
Nov 16 at 22:17
$x$ is congruent to $6$ (mod $16$) ? but not sure how that helps me? thanks
– ElMathMan
Nov 16 at 22:30
$x$ is congruent to $6$ (mod $16$) ? but not sure how that helps me? thanks
– ElMathMan
Nov 16 at 22:30
$xequiv6pmod{16}$ means $x=6+16k$ where $kinBbb Z$. So then, what is $y$?
– Lord Shark the Unknown
Nov 17 at 5:20
$xequiv6pmod{16}$ means $x=6+16k$ where $kinBbb Z$. So then, what is $y$?
– Lord Shark the Unknown
Nov 17 at 5:20
add a comment |
3 Answers
3
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up vote
1
down vote
Note that by Bezout's identity since $gcd(15,16)=1$ we have
$$15cdot (-1+kcdot 16)+16 cdot (1-kcdot 15)=1 quad kinmathbb{Z}$$
are all the solution for $15a+16b=1$ and from here just multiply by $10$.
answered Nov 16 at 22:18
gimusi
89.1k74495
add a comment |
up vote
1
down vote
In this case you don't really need the full power of the Euclidean algorithm. Since you know
$$
16 - 15 = 1
$$
you can just multiply by $10$ to conclude that
$$
16 times 10 + 15 times(-10) = 10.
$$
Now you have your $y$ and $x$.
answered Nov 16 at 22:20
Ethan Bolker
39.7k543103
wouldn't this work for $16y$+$15x$ = $10$?
– ElMathMan
Nov 16 at 22:33
It doesn't matter what you "call" $x$ and $y$. If you want $15x - 16y = 10$ just take $x = y = -10$ .
– Ethan Bolker
Nov 17 at 0:20
add a comment |
up vote
0
down vote
You have $16-15=1$
What about $$ x=-10+16k, y= -10+15k ?$$
That implies
$$ 15 x-16y=10$$
Which is a solution
answered Nov 16 at 22:35
Mohammad Riazi-Kermani
40.3k41958
add a comment |
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Note that by Bezout's identity since $gcd(15,16)=1$ we have
$$15cdot (-1+kcdot 16)+16 cdot (1-kcdot 15)=1 quad kinmathbb{Z}$$
are all the solution for $15a+16b=1$ and from here just multiply by $10$.
answered Nov 16 at 22:18
gimusi
89.1k74495
add a comment |
up vote
1
down vote
Note that by Bezout's identity since $gcd(15,16)=1$ we have
$$15cdot (-1+kcdot 16)+16 cdot (1-kcdot 15)=1 quad kinmathbb{Z}$$
are all the solution for $15a+16b=1$ and from here just multiply by $10$.
answered Nov 16 at 22:18
gimusi
89.1k74495
add a comment |
up vote
1
down vote
up vote
1
down vote
Note that by Bezout's identity since $gcd(15,16)=1$ we have
$$15cdot (-1+kcdot 16)+16 cdot (1-kcdot 15)=1 quad kinmathbb{Z}$$
are all the solution for $15a+16b=1$ and from here just multiply by $10$.
answered Nov 16 at 22:18
gimusi
89.1k74495
Note that by Bezout's identity since $gcd(15,16)=1$ we have
$$15cdot (-1+kcdot 16)+16 cdot (1-kcdot 15)=1 quad kinmathbb{Z}$$
are all the solution for $15a+16b=1$ and from here just multiply by $10$.
answered Nov 16 at 22:18
gimusi
89.1k74495
answered Nov 16 at 22:18
gimusi
89.1k74495
answered Nov 16 at 22:18
gimusi
89.1k74495
answered Nov 16 at 22:18
gimusi
89.1k74495
89.1k74495
add a comment |
add a comment |
up vote
1
down vote
In this case you don't really need the full power of the Euclidean algorithm. Since you know
$$
16 - 15 = 1
$$
you can just multiply by $10$ to conclude that
$$
16 times 10 + 15 times(-10) = 10.
$$
Now you have your $y$ and $x$.
answered Nov 16 at 22:20
Ethan Bolker
39.7k543103
wouldn't this work for $16y$+$15x$ = $10$?
– ElMathMan
Nov 16 at 22:33
It doesn't matter what you "call" $x$ and $y$. If you want $15x - 16y = 10$ just take $x = y = -10$ .
– Ethan Bolker
Nov 17 at 0:20
add a comment |
up vote
1
down vote
In this case you don't really need the full power of the Euclidean algorithm. Since you know
$$
16 - 15 = 1
$$
you can just multiply by $10$ to conclude that
$$
16 times 10 + 15 times(-10) = 10.
$$
Now you have your $y$ and $x$.
answered Nov 16 at 22:20
Ethan Bolker
39.7k543103
wouldn't this work for $16y$+$15x$ = $10$?
– ElMathMan
Nov 16 at 22:33
It doesn't matter what you "call" $x$ and $y$. If you want $15x - 16y = 10$ just take $x = y = -10$ .
– Ethan Bolker
Nov 17 at 0:20
add a comment |
up vote
1
down vote
up vote
1
down vote
In this case you don't really need the full power of the Euclidean algorithm. Since you know
$$
16 - 15 = 1
$$
you can just multiply by $10$ to conclude that
$$
16 times 10 + 15 times(-10) = 10.
$$
Now you have your $y$ and $x$.
answered Nov 16 at 22:20
Ethan Bolker
39.7k543103
In this case you don't really need the full power of the Euclidean algorithm. Since you know
$$
16 - 15 = 1
$$
you can just multiply by $10$ to conclude that
$$
16 times 10 + 15 times(-10) = 10.
$$
Now you have your $y$ and $x$.
answered Nov 16 at 22:20
Ethan Bolker
39.7k543103
answered Nov 16 at 22:20
Ethan Bolker
39.7k543103
answered Nov 16 at 22:20
Ethan Bolker
39.7k543103
answered Nov 16 at 22:20
Ethan Bolker
39.7k543103
39.7k543103
wouldn't this work for $16y$+$15x$ = $10$?
– ElMathMan
Nov 16 at 22:33
It doesn't matter what you "call" $x$ and $y$. If you want $15x - 16y = 10$ just take $x = y = -10$ .
– Ethan Bolker
Nov 17 at 0:20
add a comment |
wouldn't this work for $16y$+$15x$ = $10$?
– ElMathMan
Nov 16 at 22:33
It doesn't matter what you "call" $x$ and $y$. If you want $15x - 16y = 10$ just take $x = y = -10$ .
– Ethan Bolker
Nov 17 at 0:20
wouldn't this work for $16y$+$15x$ = $10$?
– ElMathMan
Nov 16 at 22:33
wouldn't this work for $16y$+$15x$ = $10$?
– ElMathMan
Nov 16 at 22:33
It doesn't matter what you "call" $x$ and $y$. If you want $15x - 16y = 10$ just take $x = y = -10$ .
– Ethan Bolker
Nov 17 at 0:20
It doesn't matter what you "call" $x$ and $y$. If you want $15x - 16y = 10$ just take $x = y = -10$ .
– Ethan Bolker
Nov 17 at 0:20
add a comment |
up vote
0
down vote
You have $16-15=1$
What about $$ x=-10+16k, y= -10+15k ?$$
That implies
$$ 15 x-16y=10$$
Which is a solution
answered Nov 16 at 22:35
Mohammad Riazi-Kermani
40.3k41958
add a comment |
up vote
0
down vote
You have $16-15=1$
What about $$ x=-10+16k, y= -10+15k ?$$
That implies
$$ 15 x-16y=10$$
Which is a solution
answered Nov 16 at 22:35
Mohammad Riazi-Kermani
40.3k41958
add a comment |
up vote
0
down vote
up vote
0
down vote
You have $16-15=1$
What about $$ x=-10+16k, y= -10+15k ?$$
That implies
$$ 15 x-16y=10$$
Which is a solution
answered Nov 16 at 22:35
Mohammad Riazi-Kermani
40.3k41958
You have $16-15=1$
What about $$ x=-10+16k, y= -10+15k ?$$
That implies
$$ 15 x-16y=10$$
Which is a solution
answered Nov 16 at 22:35
Mohammad Riazi-Kermani
40.3k41958
answered Nov 16 at 22:35
Mohammad Riazi-Kermani
40.3k41958
answered Nov 16 at 22:35
Mohammad Riazi-Kermani
40.3k41958
answered Nov 16 at 22:35
Mohammad Riazi-Kermani
40.3k41958
40.3k41958
add a comment |
add a comment |
Can you solve the congruence $15xequiv10pmod{16}$?
– Lord Shark the Unknown
Nov 16 at 22:17
$x$ is congruent to $6$ (mod $16$) ? but not sure how that helps me? thanks
– ElMathMan
Nov 16 at 22:30
$xequiv6pmod{16}$ means $x=6+16k$ where $kinBbb Z$. So then, what is $y$?
– Lord Shark the Unknown
Nov 17 at 5:20