★·.·´¯`·.·★ gяαρнιηg sүsтεмs σғ ιηεqυαℓιтιεs ★·.·´¯`·.·★

Salutations, fellow visitors of Mathland!

How have you all been? 

Good I hope.~

I must tell you, but I have just learned about the most fascinating piece of knowledge.  It has compelled me so much that I must share it with you all.

This fascinating thing is known as "graphing systems of inequalities".

Let us start, yes?~

In the beginning of this lesson, we must review the three equations of the three different graphs:

Line: y=mx + b

Parabola: y = (x-h)^2 + k

Circle: x^2 + y^2 = r^2

Here are the steps to graphing the equations:

1) Determine which graph is used and graph each equation.

2) Pick a test point that is not on the line.

3) Shade the plane containing the test point if the test point satisfies the equation. Shade the other plane if it does not.

Also make sure to pay attention to the signs. 

If the sign is greater or less than and equal to then the graphed line is solid. 

If the sign is only greater than or less than, then the graphed line is dashed.

Below is an example:

Well, that is all for now! See you next time.~

Goodbye.~

•´¯`•. ĊŖÄMËŖ'Ś ŖŮĻË .•´¯`•

 Greetings, fellow visitors of Mathland. 

I humbly welcome you to our newly created attraction - Cramer's Land.

In this park, we will be discussing and enjoying the simplicity and efficiency of solving for variables.

Starting off, for Cramer's Rule, there are three different variations of it.

Dx/D, Dy/D, and Dz/D

The D in the denominator represents the determinant while the Dx, Dy, and Dz, represent the matrices "row x/y/z" replaced with the solution (number after the equal sign) placed in the numerator. Once you find all the answers, you merely divide the numerator by the demons stir and you will get the answer for each variable. To check to see if you have done it right, plus the numbers back into one of the original equations and if it solves to the answer given, then you have solved the problem correctly.

Overall, using Cramer's Rule allows for one to easily find the answer to the numerical values of the digits.

I do hope you enjoyed your time here and learned a few things.~

Good day and farewell, dear guests!

׺°”˜`”°º× ѕуѕтємѕ σf єqυαтισиѕ ×º°”˜`”°º×

Hello fellow individuals of Mathland!

It has been some time since I last came upon you!

Why don't we have a short chat to reacquaint ourselves?~

I actually looked forward to introducing you all to a new topic, called System of Equations. 

Shall we begin?~

System of equations are two or three given equations with variables. In order to solve these, one must find what numbers the variables are. There are two ways in solving such a problem.

Here are 2 ways:

As you can see, both are able to be used. 

Also in solving system of equations, one has to take into consideration of the problem being either inconsistent or consistent.

Inconsistent means that there are no solutions, and if graphed, the lines would be parallel.

Consistent means that there is 1 solution (independent) or an infinite amount of solutions (dependent).

Below is an example, of a "system of equation problem" with three equations.

Well, that will be all for now! I do hope you enjoyed our discussion. 

Farewell.~

My Polar Equation Graph

ıllıllı ⓟⓞⓛⓐⓡ ⓒⓞⓞⓡⓓⓘⓝⓐⓣⓔⓢ ıllıllı

Hello, dear ladies and gentlemen of Mathland.

Today we will be having a free day among'st ourselves to explore wherever you would like to go. For those who would like to journey with me,  I will be making a delivery to the Singing Flower Garden. Along the way I will be talking about polar coordinates.

Choose wisely young ones.~

Well, lovely to have those of you who joined me for company. I give many thanks.

Now moving along.~ 

Polar equations are quite different from the usual way in graphing. 

Unlike the Cartesian Plane, the Polar Graph's shape is slightly varied. 

While the Cartesian Plane graphs points with the coordinates of (x,y) =, the polar plane graphs points using (r, θ).

Notes:

However, if one is confused, there are ways to transition between polar to rectangular and rectangular to polar coordinates. 

For polar to rectangular, one uses two equations:

x=rcos(θ)

y=rsin(θ)

For rectangular to polar:

r²=x²+y²

tan(θ)=y/x

Notes:

An important note for the theta, is that if it is greater then zero, then the graph will go counterclockwise, and if the theta is less than a zero, then the graph will go clockwise.

~~~~~~~~~~~~~~

Thank you for accompanying me on my visit to the Singing Flower garden and listening to my lecture.~

Farewell, ladies and gentlemen.

See you soon.~

—(••÷[ roтaтιng conιc ѕecтιonѕ ]÷••—

Salutations, guests of Mathland!

We will be upon our destination of the Mad Hatter's Tea Party in a few seconds!

.

.

.

I most humbly welcome you!

Today at the party we will be discussing the rotation of conic sections, otherwise known as "Axis Rotation".

Here are some notes regarding this process:

As demonstrated by the image above, in order to start, we must determine and match the coefficients with that of the first equation,  Second, we plug it into the equation and find what cot2(θ) is equal to. Next, we are able to replace the x with x' (x-prime) and y with y' (y-prime) in order to determine the coordinates on the rotated Cartesian Plane. Finally we plus it into the original equation and use algebra to simplify it.

Here is an example:

That is all! 

I do hope you enjoyed your time and our lovely conversation at the tea party! 

The March Hare seems to have enjoyed our company, for he tells us to visit again soon.

Farewell for now!

See you next time, lovely guests.~

••.•´¯`•.•• ραяαвσℓαѕ ••.•´¯`•.••

Greetings, visitors of Mathland.~

How have you all been? I have been quite lovely myself. Today, I will be taking you all on a tour through the woods to go visit the March Hare at the Mad Hatter's tea party. Along the way, I will be reviewing the concepts of Parabolas and its equations with you all.

Starting off, the base equation for parabolas is:

(x-h)_­­­_­² = 4c (y-k)

Therefore:

Vertex: (h,k)

Focus: (h, k+c)

Directrix: y=k-c

A.O.S: x=h

In order to find the "c" value in this equation, one merely sets the number in the place of "4c" equal to '4c' and solve for "c". If the "c" is greater then zero, then the graph is facing upward. However, if the "c" is less then zero, then the graph is facing downwards.

Notes:

On the other hand, if the graph is opening to the left or right, the equation switches. It changes to:

(y-kh)_­­­_­² = 4c (x-h)

Meaning:

Vertex: (h,k)

Focus: (h+c, k)

Directrix: y=h-c

A.O.S: y=k

Notes:

To find the "c" of this equation, it is still the same exact process as finding the "c" of the other equation. Only in this situation, if c is greater then zero, then the graph opens right. I fc is less then zero, then the graph opens left.

That will be all for today!~ 

We will soon arrive at the Tea Party.~

See you next time.~