if y² + xy - 3x = 9, and dy/dt = 2 when x = 2 and y = -5, what is dx/dt when x = 2 and y = -5? dx/dt =

Explanation:

Step1: Differentiate both sides with respect to $t$

Using the chain - rule and product - rule. The derivative of $y^{2}$ with respect to $t$ is $2y\frac{dy}{dt}$, the derivative of $xy$ with respect to $t$ is $x\frac{dy}{dt}+y\frac{dx}{dt}$ and the derivative of $3x$ with respect to $t$ is $3\frac{dx}{dt}$, and the derivative of 9 with respect to $t$ is 0. So, $2y\frac{dy}{dt}+x\frac{dy}{dt}+y\frac{dx}{dt}-3\frac{dx}{dt}=0$.

Step2: Rearrange the terms to solve for $\frac{dx}{dt}$

Factor out $\frac{dx}{dt}$: $\frac{dx}{dt}(y - 3)=-(2y + x)\frac{dy}{dt}$. Then $\frac{dx}{dt}=\frac{-(2y + x)\frac{dy}{dt}}{y - 3}$.

Step3: Substitute the given values

Substitute $x = 2$, $y=-5$ and $\frac{dy}{dt}=2$ into the formula for $\frac{dx}{dt}$. We have $\frac{dx}{dt}=\frac{-[2\times(-5)+2]\times2}{-5 - 3}$. First, calculate the numerator: $2\times(-5)+2=-10 + 2=-8$, then $-( - 8)\times2 = 16$. The denominator is $-8$. So, $\frac{dx}{dt}=\frac{16}{-8}=-2$.

Answer:

$-2$