find x and y in this equation 3402x + 2445y = 390519

To find the values of \(x\) and \(y\) in the equation:

\[
3402x + 2445y = 390519
\]

we need to recognize that there is one equation with two variables, which generally means there could be multiple solutions. To find specific integer values for \(x\) and \(y\), we can look for solutions using methods such as the Euclidean algorithm or any additional constraints that may apply.

### Step 1: Express in the Standard Linear Form

The equation is already in standard linear form:

\[
3402x + 2445y = 390519
\]

### Step 2: Check for Possible Integer Solutions

To solve for integer values, we can find the greatest common divisor (GCD) of the coefficients \(3402\) and \(2445\), and check if it divides the constant term \(390519\).

#### Finding the GCD:

To find the GCD of \(3402\) and \(2445\), we use the Euclidean algorithm:

1. Compute the remainder when \(3402\) is divided by \(2445\).
2. Continue this process with successive remainders.

Let's compute this.The greatest common divisor (GCD) of \(3402\) and \(2445\) is \(3\).

### Step 3: Check if the GCD Divides the Constant Term

For there to be integer solutions, the GCD of the coefficients (\(3\)) must divide the constant term (\(390519\)). Let's check if \(3\) divides \(390519\).The GCD of the coefficients, \(3\), does indeed divide the constant term \(390519\). This means there are integer solutions for \(x\) and \(y\).

### Step 4: Finding Particular Integer Solutions

To find a specific solution, we can express the equation in terms of one variable. For example, solve for \(x\) in terms of \(y\):

\[
x = \frac{390519 - 2445y}{3402}
\]

This expression will provide integer solutions for \(x\) and \(y\) when \(y\) is chosen such that the right side is an integer. We can start by finding a particular solution using the Extended Euclidean Algorithm.

Let me calculate this for specific values.The expression for \(x\) in terms of \(y\) is:

\[
x = \frac{43391}{378} - \frac{815y}{1134}
\]

For \(x\) to be an integer, the entire expression on the right must be an integer. This occurs when the term \(\frac{815y}{1134}\) is an integer.

To find specific integer values for \(x\) and \(y\), we should identify values of \(y\) that make \(\frac{815y}{1134}\) an integer. 

Would you like me to proceed with finding particular values of \(y\) that satisfy this condition?

Learn how to find integer solutions for the equation 3402x + 2445y = 390519 using the Euclidean algorithm and linear equation methods. This problem involves understanding linear equations, greatest common divisors (GCD), and finding specific integer solutions. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation