We are given the equation $y^2 + xy - 3x = 17$, along with information that $\frac{dy}{dt} = 3$ when $x = 1$ and $y = -5$. The task is to find $\frac{dx}{dt}$ under these same conditions.

Step-by-Step Process:

1. Differentiate both sides of the equation with respect to $t$: $\frac{d}{dt}(y^2 + xy - 3x) = \frac{d}{dt}(17)$ Using the chain rule and product rule, we get: $2y \frac{dy}{dt} + \left( x \frac{dy}{dt} + y \frac{dx}{dt} \right) - 3 \frac{dx}{dt} = 0$ Simplifying, this becomes: $2y \frac{dy}{dt} + x \frac{dy}{dt} + y \frac{dx}{dt} - 3 \frac{dx}{dt} = 0$

2. Substitute the given values: We are given $x = 1$, $y = -5$, and $\frac{dy}{dt} = 3$. Substituting these into the differentiated equation: $2(-5)(3) + (1)(3) + (-5) \frac{dx}{dt} - 3 \frac{dx}{dt} = 0$ Simplifying the constants: $-30 + 3 - 5 \frac{dx}{dt} - 3 \frac{dx}{dt} = 0$ This simplifies to: $-27 - 8 \frac{dx}{dt} = 0$

3. Solve for $\frac{dx}{dt}$: $-8 \frac{dx}{dt} = 27$ $\frac{dx}{dt} = -\frac{27}{8}$

Thus, the value of $\frac{dx}{dt}$ is $-\frac{27}{8}$.

Would you like more details on any of the steps or have any further questions?

Related Questions:

1. How does the product rule apply when differentiating terms like $xy$?
2. What does it mean physically if $\frac{dx}{dt}$ is negative?
3. How can you verify if the differentiation was done correctly in this problem?
4. How would this problem change if $\frac{dy}{dt}$ were a different value?
5. Could you solve this problem using implicit differentiation?

Tip:

When solving related rates problems, always ensure that every variable dependent on time is differentiated with respect to time, using the chain rule!