Let's solve the equation \( 3x - 5y = 7x + 2y \) for \(x\) and \(y\).

### Step 1: Move all terms involving \(x\) to one side of the equation and all terms involving \(y\) to the other side.
\[
3x - 7x = 2y + 5y
\]

### Step 2: Simplify the equation.
\[
-4x = 7y
\]

### Step 3: Solve for \(y\) in terms of \(x\).
\[
y = \frac{-4x}{7}
\]

So, the relationship between \(x\) and \(y\) is \(y = \frac{-4x}{7}\).

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

Here are 5 related questions to explore:
1. What is the solution to the equation if \(x = 7\)?
2. How would the solution change if the equation was \(3x + 5y = 7x + 2y\)?
3. Can you graph the equation \(y = \frac{-4x}{7}\)?
4. What happens to the relationship if you double both sides of the original equation?
5. How do you find the intersection of the line \(3x - 5y\) with another line?

**Tip:** When solving equations, always group like terms together to simplify the problem.

Learn how to solve the equation 3x - 5y = 7x + 2y with a detailed step-by-step solution. This problem covers key algebraic concepts, including equations and linear equations. Suitable for students in Grades 7-9.

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