Find all positive integer solutions for 13x + 9y = 190 using Diophantine equations.

here is the same kind of question, perhaps it helps

http://www.jiskha.com/display.cgi?id=1317362332

1--13x + 9y = 190

2--Dividing by the lowest coefficient
...y + x + 4x/9 = 21 + 1/9
3--(4x - 1)/9 must be an integer.
4--Needing a unit coefficient
...(28x - 7)/9 = 3x + x/9 - 7/9
5--(x - 7)/9 must be an integer k making x = 9k + 7
6--Substituting back into (1) yields
---y = 11 - 13k
7--k must be '0' making x = 7 and y = 11
8--Checking, 13(7) + 9(11) = 91 + 99 = 190.

To find all positive integer solutions for the equation 13x + 9y = 190, we can use Diophantine equations. A Diophantine equation is a type of equation where we need to find integer solutions.

To solve this equation, we need to use a method called the extended Euclidean algorithm. Here's how to do it step by step:

Step 1: Write the equation in the form of ax + by = c, where a, b, and c are integers. In this case, the equation is 13x + 9y = 190.

Step 2: Find the greatest common divisor (GCD) of the coefficients of x and y. In this case, the GCD of 13 and 9 is 1, which means there is a solution.

Step 3: Use the extended Euclidean algorithm to find the particular solution for x and y. The extended Euclidean algorithm involves back substitution and dividing the coefficients.

Starting with the equation 13x + 9y = 1, we can find solutions for x and y by working backward. We start with the equation 1 = 13 - 1 * 9. Then, we substitute 1 from the left side into the right side of the equation. So, we have 1 = 13 - 1 * (190 - 13x). Simplifying further, we get 1 = 13 - 190 + 13x, which can be rewritten as 189 = 13x - 12.

Now, we can see that x = 15 is a solution to this equation since 189 = 13 * 15 - 12. Substituting x = 15 into the original equation gives us 13 * 15 + 9y = 190. Solving for y, we get y = 5.

Therefore, the particular solution to the equation 13x + 9y = 190 is x = 15 and y = 5.

Step 4: Finding the general solution. To find the general solution, we need to add a constant multiple of the coefficients to the particular solution. Since the GCD is 1, we can add or subtract any multiple of 9 or 13 to the particular solution.

So, the general solution for 13x + 9y = 190 is:
x = 15 + 9t, where t is any integer.
y = 5 - 13t, where t is any integer.

These equations give all positive integer solutions for 13x + 9y = 190 using the Diophantine equation method.