Question

Diophantine confusion
find solutions of 3x+4y = 7
Please, help

Answer 1

gcd (3, 4 ) =1 and 1 | 7 hence, it has solutions
4 = 3 + 1 , hence 1 = (-1) 3 + (1) 4
multiple both sides by 7, we get 7 = 3 (-7) + 4 (7)
Hence \(x_0 = -7, y_0 = 7\)
general solution is \( x = -7 + 4t\\y = 7 - 3t\)
Now, check back:
t =0 , x = -7, y = 7 , 3x + 4y = 3*(-7) + 4(7) = -21 + 28 = 7 \(\checkmark \)
t =1, x = -7 + 4 = -3 , y = 7 -3 =4, hence 3(-3) + 4*4 = 7\(\checkmark\)

Answer 2

\[3x+4y=7 \\ 4y=-3x+7 \\ y=\frac{-3}{4}x+7 \\ \text{ this says slope is } \frac{-3}{4}\]
initially we can see (1,1) as a solution... since 3+4=7
-3/4=(-3n)/(4n)
we can find the other solutions by using the slope.
remember slope is rise/run
the rise happens to the y
and the run happens to the x
\[(1+4n,1-3n)\]
n is an integer.

checking...
\[3x+4y=7 \\ 3(1+4n)+4(1-3n)=7 \\ 3+12n+4-12n=3+4=7 \text{ true!}\]

Answer 3

\[3x+4y=7 \\ 3(-7+4t)+4(7-3t)=-21+12t+28-12t=7 \text{ your solution also works }\]

Answer 4

we just used a different initial point

Answer 5

so, we have many general solutions for the problem, right?

Answer 6

there is not one way to express it

Answer 7

so yes

Answer 8

you just find a point
and use the slope to find all the integer points that work for 3x+4y=7

Answer 9

oh, I never know about it. Thank you so much. It scared me a lot.

Answer 10

I can't. When making question here, I tried to understand the concept in different ways. However, I lost a lot of credits when I applied the better way here on my solution. I asked my Prof the reason why I didn't get credit when I was not wrong. He answered: Your solution is not wrong but what is the goal of the lecture??? and I...... dumb.!!

Answer 11

\[ax+by=c \\ \text{ say you see the integer pair } (m,n) \text{ works } \\ \text{ then you can state the answer as } \\ (m+tb,n-ta) \\ \text{ or could be expressed as } \\ (m-tb,n+ta) \\ \text{ depending if you read the slope as } \frac{-a}{b} \text{ or as } \frac{a}{-b}\]

Answer 12

what he took off points on your solution you posted here?

Answer 13

No, I have to follow his way on solving problems. If I got right but not follow his way, my credit is off. That is why most of the time, when I get stuck or confuse, I asked for checking, not new way.

Answer 14

the weird thing about this diophantine equations is I think people get confused about them in number theory..even though these questions are more algebraic...Like all you have to do is remember what a line is and what the slope of a line is.

Answer 15

Question: how to type "not divide" in latex?

Answer 16

\[\cancel{|}\]

Answer 17

cancel{ | }

Answer 18

Thank you.
Oh, about diophantine, since the solutions must be integers, do we have to restrict t when your b / d is a fraction?

Answer 19

t is an integer

Answer 20

also a,b,c are integers too

Answer 21

in my thingy above

Answer 22

Not the case, I saw my craziness!! (a, b) = d, there is no way to have \(d\cancel{|} b\)

Answer 23

Again, thanks a lot @freckles

Answer 24

np

Answer 25

Here is one more example just in case:

5x+9y=4

I know 9-5 is 4
so one solution could be (-1,1)
And remember we can find the slope of this line which will help give us all the integer pairs that satisfy the line

\[9y=-5x+4 \\ y=\frac{-5}{9} x+\frac{4}{9} \\ \text{ the slope is } \frac{-5}{9} (=\frac{rise}{run}) \\ \text{ the integer pairs that satisfy the line is } (-1+9t,1-5t) \\ \text{ where } t \text{ is an integer }\]

Answer 26

though you should pick your initial point on however the teacher wants you too since he seems to be picky

Answer 27

Got you, I love solving problem in many ways like this. I understand it more.

Answer 28

this one should be the most easiest to visualize
x+y=1
an obvious solution is (1,0)