We are tasked with solving the equation:

$47x + 43y = \gcd(47, 43) = 1$

This is a linear Diophantine equation where the coefficients $a = 47$, $b = 43$, and the constant $c = \gcd(47, 43) = 1$.

We can solve it using the Extended Euclidean Algorithm to find integers $x$ and $y$.

Step 1: Use the Euclidean Algorithm to find $\gcd(47, 43)$

The Euclidean Algorithm involves repeated division to find the remainder:

1. $47 \div 43$ gives quotient $1$ and remainder:

   $47 - 1 \cdot 43 = 4$

2. Next, $43 \div 4$ gives quotient $10$ and remainder:

   $43 - 10 \cdot 4 = 3$

3. Then, $4 \div 3$ gives quotient $1$ and remainder:

   $4 - 1 \cdot 3 = 1$

4. Finally, $3 \div 1$ gives remainder $0$, so:

   $\gcd(47, 43) = 1$

Now we backtrack to express $1$ as a linear combination of $47$ and $43$.

Step 2: Back-substitution to express $1$ as $47x + 43y$

We reverse the steps of the Euclidean Algorithm:

1. From $4 - 1 \cdot 3 = 1$, we write:

   $1 = 4 - 1 \cdot 3$

2. Substitute $3$ using $43 - 10 \cdot 4$ (from Step 2 of the Euclidean Algorithm):

   $3 = 43 - 10 \cdot 4$

   Substituting into $1 = 4 - 1 \cdot 3$, we get:

   $1 = 4 - 1 \cdot (43 - 10 \cdot 4)$

   Simplify:

   $1 = 4 - 1 \cdot 43 + 10 \cdot 4$

   Combine terms involving $4$:

   $1 = 11 \cdot 4 - 1 \cdot 43$

3. Substitute $4$ using $47 - 1 \cdot 43$ (from Step 1 of the Euclidean Algorithm):

   $4 = 47 - 1 \cdot 43$

   Substituting into $1 = 11 \cdot 4 - 1 \cdot 43$, we get:

   $1 = 11 \cdot (47 - 1 \cdot 43) - 1 \cdot 43$

   Simplify:

   $1 = 11 \cdot 47 - 11 \cdot 43 - 1 \cdot 43$

   Combine terms involving $43$:

   $1 = 11 \cdot 47 - 12 \cdot 43$

Step 3: Identify the solution

We have expressed $1$ as:

$1 = 11 \cdot 47 - 12 \cdot 43$

Thus, $x = 11$ and $y = -12$ is a solution.

Final Answer:

The pair $(x, y)$ is:

$(x, y) = (11, -12)$

Let me know if you need further clarification or additional examples!

Related Questions:

1. What are the steps of the Extended Euclidean Algorithm?
2. How can we generalize solutions for linear Diophantine equations $ax + by = c$?
3. What happens if the gcd is not $1$?
4. Can we verify the solution $47x + 43y = 1$ directly?
5. How can we find additional solutions to this equation?

Tip:

In linear Diophantine equations, if $\gcd(a, b) = d$, solutions exist only if $d$ divides $c$. Always use the Extended Euclidean Algorithm for such problems.