The given equation is:

\[
15X + 12Y = 141
\]

This is a linear Diophantine equation, which means that we're typically looking for integer solutions for \( X \) and \( Y \). However, we can also express one variable in terms of the other to find multiple solutions.

### Solving for \( X \) in terms of \( Y \):

1. First, simplify the equation by dividing through by the greatest common divisor (GCD) of 15 and 12, which is 3:

\[
5X + 4Y = 47
\]

2. Now, solve for \( X \) in terms of \( Y \):

\[
5X = 47 - 4Y
\]
\[
X = \frac{47 - 4Y}{5}
\]

For \( X \) to be an integer, \( 47 - 4Y \) must be divisible by 5. So, we solve the congruence:

\[
47 - 4Y \equiv 0 \pmod{5}
\]
\[
47 \equiv 2 \pmod{5}
\]
\[
4Y \equiv 2 \pmod{5}
\]

Next, solve for \( Y \). The inverse of 4 modulo 5 is 4 (since \( 4 \times 4 \equiv 1 \pmod{5} \)), so:

\[
Y \equiv 2 \times 4 \pmod{5}
\]
\[
Y \equiv 8 \pmod{5}
\]
\[
Y \equiv 3 \pmod{5}
\]

Thus, \( Y = 5k + 3 \) for some integer \( k \).

3. Substitute \( Y = 5k + 3 \) into the equation for \( X \):

\[
X = \frac{47 - 4(5k + 3)}{5}
\]
\[
X = \frac{47 - 20k - 12}{5}
\]
\[
X = \frac{35 - 20k}{5}
\]
\[
X = 7 - 4k
\]

### General Solution:

The solutions for \( X \) and \( Y \) are:

\[
X = 7 - 4k
\]
\[
Y = 5k + 3
\]

where \( k \) is any integer.

### Example Solutions:
- For \( k = 0 \): \( X = 7 \), \( Y = 3 \)
- For \( k = 1 \): \( X = 3 \), \( Y = 8 \)
- For \( k = -1 \): \( X = 11 \), \( Y = -2 \)

Would you like further details or have any questions about this solution?

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Here are 5 questions to deepen your understanding:
1. What is a Diophantine equation, and why is it significant?
2. How do we find the modular inverse in equations like this?
3. Can we solve \( 5X + 4Y = 47 \) graphically to find integer solutions?
4. What is the role of the GCD when simplifying linear equations?
5. How do these types of equations apply to real-world problems?

**Tip:** When solving Diophantine equations, always check if the GCD of the coefficients divides the constant term; this determines if integer solutions exist.

Discover how to solve the linear Diophantine equation 15X + 12Y = 141 using algebraic and modular arithmetic techniques. This step-by-step guide simplifies the problem and provides integer solutions. Suitable for students in Grades 9-12.

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