find `dy/dx`
`5x^2+5y^2-7y+3x=2` ,

Answer

To find \(\frac{dy}{dx}\) for the equation \(5x^2 + 5y^2 - 7y + 3x = 2\), we will differentiate both sides of the equation with respect to \(x\).

### Step-by-Step Solution:

1. **Differentiate both sides of the equation:**
\[ \frac{d}{dx}(5x^2 + 5y^2 - 7y + 3x) = \frac{d}{dx}(2) \]
The right side becomes \(0\) since the derivative of a constant is \(0\).

2. **Apply the differentiation rules:**
- The derivative of \(5x^2\) is \(10x\).
- The derivative of \(5y^2\) is \(10y \frac{dy}{dx}\) (using the chain rule).
- The derivative of \(-7y\) is \(-7 \frac{dy}{dx}\).
- The derivative of \(3x\) is \(3\).

Putting it all together, we have:
\[ 10x + 10y \frac{dy}{dx} - 7 \frac{dy}{dx} + 3 = 0 \]

3. **Rearranging the equation:**
Group the terms involving \(\frac{dy}{dx}\):
\[ 10y \frac{dy}{dx} - 7 \frac{dy}{dx} + 10x + 3 = 0 \]
This can be rewritten as:
\[ (10y - 7) \frac{dy}{dx} + 10x + 3 = 0 \]

4. **Isolate \(\frac{dy}{dx}\):**
\[ (10y - 7) \frac{dy}{dx} = - (10x + 3) \]
Now, divide both sides by \(10y - 7\):
\[ \frac{dy}{dx} = \frac{- (10x + 3)}{10y - 7} \]

5. **Simplify the expression:**
We can factor out a negative sign:
\[ \frac{dy}{dx} = \frac{10x + 3}{7 - 10y} \]

Thus, the final result is:
\[ \frac{dy}{dx} = \frac{10x + 3}{7 - 10y} \]