Using the Euclidean Algorithm to Find Values for x and y in Linear Combinations?

[Caldus]

I need to be able to plug in appropriate x and y values for:

154x + 260y = 4

I guess this is done by working the euclidean algorithm backwards. But how do you do that?

 

[jamesrc]

Well, it's been like forever and a day since I did problems like this, but I think it goes something like this:

154x + 260y = 4

260 = (1)*154 + 106
154 = (1)*106 + 48
106 = (2)*48 + 10
48 = (4)*10 + 8
10 = (1)*8 + 2
8 = (4)*2

4 = 2*2 = 2*(10-8) = 2*(10-(48-4*10))
= 10*10 - 2*48
= 10*(106 - 2*48) - 2*48
= 10*106 - 22*48
= 10*(260 - 154) - 22*(154-106)
= 10*260 - 32*154 + 22*106
= 10*260 - 32*154 + 22*(260-154)
= 32*260 - 54*154

so x = -54 and y = 32

 

[tacman]

To find appropriate values for x and y in the equation 154x + 260y = 4, we can use the Euclidean algorithm to work backwards. This algorithm involves finding the greatest common divisor (GCD) of the two coefficients, 154 and 260. The GCD of 154 and 260 is 2.

Next, we can divide both sides of the equation by the GCD to simplify it and get rid of any common factors. This gives us 77x + 130y = 2.

Now, we need to express the GCD, 2, as a linear combination of 77 and 130. This can be done by using the extended Euclidean algorithm. The extended Euclidean algorithm involves finding the coefficients, s and t, such that the GCD can be expressed as s(77) + t(130).

In this case, the extended Euclidean algorithm gives us s = -1 and t = 1. This means that we can express 2 as -1(77) + 1(130).

Substituting this value for 2 into our simplified equation, we get -1(77x) + 1(130y) = -1(77x + 130y) = -1(2) = -2.

Therefore, we can plug in x = -1 and y = 1 to satisfy the original equation of 154x + 260y = 4. This is just one possible solution, as there are infinitely many solutions to this equation. Other values of x and y that satisfy the equation include x = -3 and y = 2, x = 4 and y = -3, etc.

In summary, to find appropriate values for x and y in a Euclidean linear combination, we use the Euclidean algorithm to find the GCD, then express the GCD as a linear combination of the coefficients in the equation, and finally plug in the values for s and t to get a solution.