Cramers rule

Cramer's rule is another way to solve a system of equations without using substitution or elimination.  The equation for cramer's rule is x=Dx/d y=Dy/d z=Dz/d.  To use this equation you have to first find D which is the determinant of the equation.  To find x you substitute the answers into the x slot and then find the determinant and divide that by D.  You do this to all of the variables to find them.  If D=0 then the equation is inconsistent and has no answers.  Here is an example of Cramer's rule. 

Systems of equations

Systems of Equations are two equations that need to be solved together.  There are two types of answers to the equations; inconsistent or consistent.  Inconsistent equations have no solution and are usually parallel lines.  Consistent equations have solutions and if there is only 1 solution it is independent and if there are infinite solutions than it is dependent.  You can solve these equations through either substitution or elimination.  With substitution you first solve 1 equation for 1 variable and then substitute that into the other equation.   Then you solve for the variable and substitute back to find the 2nd variable. With elimination you interchange any two equations and multiply by a constant.  Then you add one equation to the other to eliminate a variable. Here is an example of this kind of problem. 

Polar coordinates

Polar coordinates are similar to coordinates on a unit circle but a little different.  To graph polar coordinates you need to use r and theta.  Theta is the angle of the coordinate and r is equal to the radius.  There are four different ways to graph polar coordinates with a positive r and theta, negative r and theta, positive r negative theta, negative r positive theta.  You can also convert rectangular points to polar points using  the formula r^2=x^2+y^2; tantheta=y/x.  To convert polar coordinates to rectangular use the formula x=rcostheta y=rsintheta.  Here is a picture of a polar coordinate graph.  

Graphs of polar equations

There are several different graphs and equations for the polar graph.  For circles at the origin the equation is r=a and |a|=radius and the equation for spirals is r=atheta.  For circles with a center on an axis the equations are r=asintheta; r=acostheta and a=diameter.  The equations for cardiods are r=a+asintheta; r=a-sintheta; r=a+acostheta; r=a-costheta. Limacons have a similar equation with it being r=a+bsintheta; r=a-bsintheta; r=a+bcostheta; r=a-bcostheta when a/b is less than 1.  The equation for rose curves are r=asin(n)theta and r=acos(n)theta and an odd n means n is the number of petal while an even n means that 2n equals the number of petals.  Lastly the equations for lemniscates are r^2=acos2theta and r^2=asin2theta.  Here is a picture of some of the graphs of polar equations. 

Rotating Conics

To rotate comics you first need to make sure your equation is in the right form. This form is Ax^2+Bxy+Cy^2+Dx+Ey+F=0. Once you do this you can find the angle with the equation cot2theta=(A-C)/B where theta is in between 0 and 90. Once you do that you can replace x and y with the equations x=x'costheta-y'sintheta and y=x'sintheta+y'costheta. After that you can use algebra to simplify. Some equations that can help with rotating conics is 1+cot^2theta=csc^2(2)theta and cot2theta=cos2theta/sin2theta. Here is a picture of a rotated conic. 

My polar equation picture

Parabolas

A parabola is a curve that is equidistant from the directrix and focus. The general equation of a vertical parabola is (x-h)^2=4c(y-k). For a horizontal parabola the general equation is (y-k)^2=4c(x-h). To graph a parabola you need to find the vertex, focus, directrix and axis of symmetry. For a vertical parabola to find the vertex you find (h,k) the focus is (h, k+c), the directrix is y=k-c and the axis of symmetry is x=h. For a horizontal parabola the vertex is also (h, k), the focus is (h+c,k) the directrix is x=h-c and the axis of symmetry is y=k. Here is an example of a parabola.