Convert the equalities below equalities by adding sack variables. Use s_1 and s_2 for your slack variables.

Use polar coordinates to find the limit. [Hint: Let ${\displaystyle x=r\cos \text{and}y=r\sin }$ , and note that (x, y) (0, 0) implies r 0.] ${\displaystyle \underset{(x,y)\to (0,0)}{lim}\frac{x^2-y^2}{\sqrt{x^2+y^2}}}$

Lynbrook West , an apartment complex , has 100 two-bedroom units. The montly profit (in dollars) realized from renting out x apartments is given by the following function.
${\displaystyle P\left(x\right)=-12x^2+2136x-41000}$
To maximize the monthly rental profit , how many units should be rented out?
What is the maximum monthly profit realizable?

First determine whether the solutions of the quadratic equation real or complex without solving the equation. Then solve the equation.
${\displaystyle 3x^2-4x-15=0}$
Determine whether the solutions of the quaratic equation are real or complex without solving the equation.

The random variables X and Y have joint density function
${\displaystyle f(x,y)=\frac{2}{3}(x+2y),0\le x\le 1,0\le y\le 1}$
Find the marginal density functions of X and Y.
${\displaystyle f_X\left(x\right)=?}$
${\displaystyle f_Y\left(y\right)=?}$
Are X and Y statistically independent random variables?

The analysis of shafts for a compressor is summarized by conformance to specifications. Suppose that a shaft is selected at random. What is the probability that the selected shaft conforms to surface finish requirements or to roundness requirements?
$$\begin{array}{|ccc|}\hline & & Roundness\\ & & Conforms& Doesnot\\ & & & Conforms\\ Surface& Conform& 345& 5\\ Finish& Doesnot& 1& 8\\ & Conform\\ \hline\end{array}$$

${\displaystyle f(x,y)=x\sin \left(x\right)+\cos \left(x\right)-y\sin \left(x\right)+\frac{y^2}{2}}$
My task is to find the critical points of this multivariable function (Determine the set of critical points of the function). Now I found the partial derivative to be${\displaystyle f_x^{\prime }=(x-y)\cos \left(x\right)}$
${\displaystyle f_y^{\prime }=y-\sin \left(x\right)}$
Wolfram Alpha says the critical point(s) is/are at ${\displaystyle x=y}$? How do I get the above in that form? The best I could do was get it in the form ${\displaystyle x-y\sec \left(x\right)+\tan \left(x\right)=y}$

Solve the following equations. Give exact simplified values and decimal approximations to three decimal places , as appropriate. Solutions may be complex , and complex solutions must be written in standard form.
$x(2-2x)=3$