Let's solve the given Linear Diophantine Equations for integer solutions.

1. $4x + 3y = 7$

We need to find integer values of $x$ and $y$ that satisfy this equation.

First, we can express $y$ in terms of $x$:

$y = \frac{7 - 4x}{3}$

For $y$ to be an integer, $7 - 4x$ must be divisible by 3. Let's test integer values for $x$:

• For $x = 1$:

$y = \frac{7 - 4(1)}{3} = \frac{7 - 4}{3} = \frac{3}{3} = 1$

So, $x = 1$ and $y = 1$ is a solution.

Solution: $x = 1, y = 1$ is an integer solution.

2. $15x + 25y = 10$

This equation can be simplified by dividing the entire equation by 5:

$3x + 5y = 2$

Now, express $y$ in terms of $x$:

$y = \frac{2 - 3x}{5}$

For $y$ to be an integer, $2 - 3x$ must be divisible by 5. Let's test integer values for $x$:

• For $x = 4$:

$y = \frac{2 - 3(4)}{5} = \frac{2 - 12}{5} = \frac{-10}{5} = -2$

So, $x = 4$ and $y = -2$ is a solution.

Solution: $x = 4, y = -2$ is an integer solution.

3. $6x + 9y = 3s$

We need to find integer values of $x$, $y$, and $s$ that satisfy this equation.

We can simplify the equation by dividing the entire equation by 3:

$2x + 3y = s$

Now, $s$ can be any integer that satisfies this equation. We can choose $s$ such that $x$ and $y$ are integers.

Let's express $y$ in terms of $x$ and $s$:

$y = \frac{s - 2x}{3}$

For $y$ to be an integer, $s - 2x$ must be divisible by 3. Since $s$ can be any integer, this equation can have infinite integer solutions depending on the choice of $s$.

For example, if $s = 1$:

$y = \frac{1 - 2x}{3}$

Testing $x = 1$:

$y = \frac{1 - 2(1)}{3} = \frac{1 - 2}{3} = \frac{-1}{3} \quad \text{(not an integer, so not a solution)}$

Testing $x = 2$:

$y = \frac{1 - 2(2)}{3} = \frac{1 - 4}{3} = \frac{-3}{3} = -1$

So, $x = 2$, $y = -1$, and $s = 1$ is a solution.

Solution: One solution is $x = 2$, $y = -1$, $s = 1$.

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Do you need more details or have any other questions?

Related Problems:

1. Find the integer solutions of $7x + 5y = 10$.
2. Solve the Diophantine equation $2x + 3y = 1$.
3. Determine the integer solution of $8x + 6y = 14$.
4. Find all integer solutions of $9x + 4y = 20$.
5. Solve for integer solutions $5x + 7y = 12$.

Tip: For Linear Diophantine Equations of the form $ax + by = c$, always check if the greatest common divisor (GCD) of $a$ and $b$ divides $c$. If not, there are no integer solutions.