A Linear Diophantine Equation | 3x+4y=17

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[musicramblingsbyeben]

My approach would be:
3x = 17-4y
for the right side to be positive, y<5, then plug-in values of y from 1 up to 4 and you will get same results. But this only applies if you consider only positive integers as solution, otherwise you need to use the Euclidean algorithm.

[hugh]

By the Extended Euclidean Algorithm we get 3*3+4*(-2)=1. Multiply by 17 to get 3*51+4+(-34)=17. Then we can keep taking 4 from 51 and adding 3 to -34 until we get 3 and 2, for 3*3+4*2=17. I wrote a very bad Excel Macro for this back in the day

[ayushshinde]

can be done like this
3x+4y=3x+(3+1)y=17
3(x+y)+y=17; let x+y=m, y=n, so m≥n
minimum value of m is when m=n
3m+m=17, so m=4.25, m≥4.25 and also m<6 cuz otherwise it will exceed 17
so m has values between 4.25 and 6 that is 5
so x+y=5, y=2
(x, y)=(3, 2)

[eugenemasoniv]

Very interesting approach! I like your videos because they're not just about "finding x" but utilizing different ways of coming to a solution. Interestingly enough, this inspired me to produce the following:

First, rearrange the equation 3x+4y=17 to 3x-1=4(4-y). Notice that if we set x to the smallest positive integer, 1, we get 2, another positive integer. Thus, y≤3 because if y>3, 4(4-y) is a non-positive integer.

Also notice that if we set y to 1, we get 12, a multiple of 3. However, 3x-1 is one away from a multiple of 3. Thus y is either 2 or 3.

4(4-y) is even so 3x-1 must be even. It must also be a multiple of 4. However, when x is even, 3x-1 is odd. So x must be odd.

Now rearrange the equation to x=[4(4-y)+1]/3. If we set y to 2, we get x=[4(4-2)+1]/3=3. And 3 is an odd integer, what x should be. However, if we set y to 1, x=[4(4-1)+1]/3=13/3. Thus x=3 and y =2.

[zaj]

My quick and dirty approach was like this:
x has to be odd (Otherwise the sum wouldn't be odd).
x=1 doesn't work,
x=3 does work,
x=5 doesn't work.
(For x>5 the sum is greater than 17.)

[hamzanazim]

3x + 4y = 17
x = (17-4y)/3
Therefore 17-4y should be integral multiple of 3 and so 17-4y belongs to (0, 3, 6, 9, 12, 15) out of these only 9 give integral answer for y
x = 3, y = 2

[abiymamo]

if 2a:3b=5:6 and 3b:2c=36:15 then what is a:c please do it❤❤

[mircoceccarelli]

3x + 4y = 17

x = 3 + 4k, y = 2 - 3k, k intero .

( k, x, y ) = ( 0, 3, 2 ), unica soluzione
in cui i valori sono tutti positivi !

[wiikoolaid]

Whats diophantine? Never heard of this stuff