PLEASE SHOW DETAIL to example 51. WILL RATE LIFESAVER if done so!! In class, it

Question

PLEASE SHOW DETAIL to example 51. WILL RATE LIFESAVER if done so!!

In class, it was claimed that there are infinitely many different pairs of integers (x, y ) with the property that 3x + 2y = ged(3, 2) = 1. For convenience, let's call such a pair of integers (x, y) a solution to the equation 3x + 2y = 1. How did we find the first such solution? How did we find infinitely many other solutions? Describe the solutions we found. (Notice our 1-2 punch: find one, then construct more.) A key question we would like to know the answer to is this question: Does our method of finding infinitely many solutions actually find ALL of the solutions (x. y) to the equation 3x + 2y = 1? Or are there others? What do you think? How can you know for sure?

Explanation / Answer

since 3 and 2 are co prime

3x + 2y = 1

by in spection x = 1 and   y = -1

x = 1 + 2t   and y = -1 - 3t for any value of t

thsu

x = 1+2t and y = -1 -3t   forms so many solutiosn

some of them are

for t = 0   x =1 and y = -1

t =1 x = 3 and y = -4

t=2 x = 5 and y = -7

thus ans

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