four solution x+3y=11
Answers
Simply, give a value either for x or for y. I'm giving a value for x.
Let x = 2. So we get y = 3.
2 + 3 × 3 = 2 + 9 = 11
Now we get that the ordered pairs (x, y) = (2 + 3k, 3 - k) for any integer k.
We have an ordered pair of solution (x, y) = (2, 3).
k = 1 implies (x, y) = (5, 2).
k = 2 implies (x, y) = (8, 1).
k = 3 implies (x, y) = (11, 0).
Why increasing order?!
k = -1 implies (x, y) = (-1, 4).
k = -2 implies (x, y) = (-4, 5).
k = -3 implies (x, y) = (-7, 6).
So on...!
If you feel trouble guessing a solution, see the following.
We have the equation x + 3y = 11. Here I'm replacing RHS by 1, hence we get a new equation,
x + 3y = 1
We will multiply both sides by 11 once a solution is got. Here we're applying Euclid's Division Lemma for the solutions.
EDL is applied relating the coefficients of x and y, here they're 1 and 3 respectively.
So,
3 = 1 × 2 + 1
From this we can write that,
1 = 3 - 2
1 = - 2 + 3 × 1
Now we multiplying both sides by 11, so,
11 = - 2 × 11 + 3 × 1 × 11
11 = - 22 + 3 × 11
This implies (x, y) = (- 22, 11).
So finally we get (x, y) = (- 22 + 3k, 11 - k) for any integer k.
From (x, y) = (2 + 3k, 3 - k), we get the pair (- 22, 11) when k = - 8.