Convert between the coordinate systems. Use the conversion formulas and show work. Spherical: (8,frac{pi}{3}, frac{pi}{6}) Change to cylindrical.

To find: The equivalent polar equation for the given rectangular-coordinate equation.
Given:
${\displaystyle x=r\cos \theta }$
${\displaystyle y=r\sin \theta }$
b. From rectangular to polar:
${\displaystyle r=\pm \sqrt{x^2+y^2}}$
${\displaystyle \cos \theta =\frac{x}{r},\sin \theta =\frac{y}{r},\tan \theta =\frac{x}{y}}$
Calculation:
Given: equation in rectangular-coordinate is ${\displaystyle y=x}$.
Converting into equivalent polar equation -
${\displaystyle y=x}$
Put ${\displaystyle x=r\cos \theta ,y=r\sin \theta ,}$
${\displaystyle \Rightarrow r\sin \theta =r\cos \theta }$
${\displaystyle \Rightarrow \frac{\sin \theta }{\cos \theta }=1}$
${\displaystyle \Rightarrow \tan \theta =1}$
Thus, desired equivalent polar equation would be ${\displaystyle \theta =1}$

As you can see the solution to the system is the coordinates of the point where the two lines intersect. So, when solving linear systems with two variables we are really asking where the two lineas will intersect.

The systems used to find the location of the point in a plane and find the rectangular and polar coordinates for the point P.

Consider the following two bases for ${\displaystyle R^3}$ :
$\alpha :=\{\left[\begin{array}{c}2\\ 1\\ 3\end{array}\right],\left[\begin{array}{c}-1\\ 0\\ 1\end{array}\right],\left[\begin{array}{c}3\\ 1\\ -1\end{array}\right]\}and\beta :=\{\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right],\left[\begin{array}{c}-2\\ 3\\ 1\end{array}\right],\left[\begin{array}{c}2\\ 3\\ -1\end{array}\right]\}$ If $[x{]}_{\alpha }={\left[\begin{array}{c}1\\ 2\\ -1\end{array}\right]}_{\alpha }thenfind[x{]}_{\beta }$
(that is, express x in the ${\displaystyle \beta }$ coordinates).

Solving Systems Graphically- One Solution
Find or create an example of a system of equations with one solution.
Graph and label the lines on a coordinate plane. Provide their equations.
State the accurate solution to the system.

To find: the distance between the lakes to the nearest mile.
Given:
(26,15) and (9,20)

The equivalent polar equation for the given rectangular-coordinate equation:
${\displaystyle x^2+y^2+8x=0}$