HI MISS V

I FINISHED I FINISHED I FINISHED I FINISHED

IM D O N E DONEEE

NOW I CAN BE ON SUMMER

BYE LOVE YOU

YOU'RE THE BEST

Review Presentation: Equation of a Plane

In order to find the equation of a plane, you must use the formula:
a(x-x1) + b(y-y1) + c(z-z1) + 0
When asked to find the equation of a plane, you will be given three points.  The first step is to choose one point to be your initial, one point to be u, and one point to v.  But, since they are points, you must make them into vectors by using your initial point and doing: terminal - initial.
(1,2,3)v
(-1,5,4)u
(o,-3,7)i
u = <-1,8,-3>
v = <1,5,-4>

Now you must place your vectors into a cross product (u x v) so that you can find your a,b, and c.
     i     j     k     i     j
    -1   8    -3   -1    8
     1   5    -4    1    5
u x v = <-17,-7,-3>
Now you plug these into your original equation, with x1, y1, and z1, being your points from your initial.

Final answer = -17x - 7y -3z = 0

Limits -> Infinity

When presented with limits to infinity, you are usually given a fraction that is similar to p(x)/q(x). When this occurs, there are three rules you must follow that will make your answers so much easier!

If n > m your limit is 0
If n = m your limit is the numerator/denominator
If your n > m your limit does not exist

n = power of the numerator
m = power of the denominator

For example
lim       2x+1/x^2
x -> infinity

n < m = limit = 0

Limits

Limits of Polynomials are one of the easiest things of Math Analysis EVER! All you have to do is plug in the number!
For instance

lim 2x+1
x -> 4
 = 2(4) + 1
 = 9
WINNER WINNER CHICKEN DINNER

Limits of Rational Functions are a little big more difficult, because they usually equal 0/0 which means they are intermediate!
When this occurs, you must do cancellation. This is when you factor your given function, cancel out terms, and then plug in your x again.

lim        x-1/(x-1)(x+1)
x-> 1
 = cancel out x - 1
 plug 1 in for x
 = 1/2

Review Presentation: Equation of a Sphere

When finding the center of a sphere, you will be given an equation of a sphere. And all you have to do is complete the square essentially..but three times. For instance...

x^2 + y^2 - 2x + z^2 - 4z + 10y - 4 = 0
First you should combine the terms
x^2 - 2x + _ + y^2 + 10y + _ + z^2 - 4z + _ = 4
Now complete the square three times!
(x^2 - 2x + 1) + (y^2 + 10y + 25) + (z^2 - 4z + 4) = 4 + 1 + 25 + 4
Factor
(x - 1)^2 +  + 5)^2 +  - 2)^2 = 34
Pull out terms!
center: (1, -5, 2)
radius: √34

YAY

Review Presentations: Distance Between a Point and a Plane

When finding the distance between a point and a place, you will be given a point as well as an equation of a plane.  The formula for finding distance is: D = |PQ x n|/||n||
Q will be given to you as a point, and n will be the numbers you pull out from the equation.  For example Q (1,5,2) and 2x +3y -z + 5. n = <2,3,-1>
The next step deals with finding vector PQ. You will use Q as your terminal and you will create an initial by zeroing two variables from the plane equation. This will give you P = (0,0,-5) Then you must find the dot product of PQ and n which will equal -20, but the absolute value of that is 20. and you will divide that by the magnitude of n which is found by the square root of all the numbers squared of the vector which will equal root 14.
Your final answer will be 20/√14

The Limit Does Not Exist

In chapter 12 we learned about limits. Which is when a slope or curve has a boundary. But, there are times when limits don't exist.

1. f(x) approaches different numbers from left and right

2. f(x) increases or decreases without bound

3. f(x) oscillates between two fixed values

Review Presentation: Cross Product

Cross Products involve matrices with vectors. To find the cross product you must create a matrix using I, j, and k. You will be given two vectors: u and v, and you must find the cross product by using matrices and determinants, which will form an equation.

u: <2,1,5>

V: <-3,4,2>

these are your vectors and now you must place them into a matrix.

         i     j     k     i     j     

        2    1.   5.   2.   1

       -3.   4.   2.  -3.   4

u x v: 2i -15j + 8k + 3k -20i -4j

u x v: -18i -20j + 11k

u x v: <-18, -20, 11>

Review Presentation: Parametric Equations

Parametric equations consist of the variables, t, x, and y. When doing a parametric equation, we must remember that we must create a table, a graph, and then finally eliminate the parameter.

When making a table, we must use t, x, y. We can plug in any variable as long as we include negatives as well as positives. I usually use -2, -1, 0, 1, and 2. Once we place these numbers into our table, we simply plug them into the equation which is how we will get our x and y values. The next step is to plot the points onto the graph, and placing t = (number) near the point. This is when you will get a curve. 

The last thing you must do is eliminate the parameter. In the x equation, you must solve for t. And then plug t into the y equation, which is how you will eliminate the parameter. And that's how you do it!

Review Presentations: Mathematical Induction

Rebecca presented on mathematical induction, which is basically a proof. You must attempt to make e left hand side equal to the right hand side, this is how you begin.

First, when given an equation you must prove true for n=1. This is when you plug in 1 for each value of n. the left hand side and the right hand side must be equal.

Second, you must assume true for n = 1. Then prove true for n + 1. This is when you create a new equation. The right hand side of the equation + the left hand side (with n+1) = left hand side (with n+1). The end result is that each side must be equal to each other, and that is how you do mathematical induction.

The final thing that you have to do is state that the left hand side is equal to the right hand side through mathematical induction hence true for n + 1.

3-D Graphing

Power of the Polar Equations

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What are polar equations actually used for? They are actually used for navigation, using the angle and distance from the object (center). But, they often use angle measures instead of radians in giving directions.  In addition, radial symmetry uses the polar coordinate system as the central point acting as a pole.  Hence, everything is revolved around the pole.  In addition, radially asymmetric systems are modeled with polar coordinates.  For instance, a microphone uses a pattern that illustrates the response of the sound from a certain direction.  These are usually explained through polar curves.

In addition, they are used to make fun art!!

Lesson 10.8

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Today we learned about the polar equations of conics, which includes ellipses, hyperbolas, and parabolas.  There are some very important equations you need to do in order to find the polar equations.

• r = ep/ 1+esinθ
• horizontal directrix that is above the pole (positive y)
• r = ep/1-esinθ
• horizontal directrix that is below the pole (negative y)
• r = ep/1+ecosθ
• vertical directrix that is on the right of the pole (positive x)
• r = ep/1-ecosθ
• vertical directrix that is on the left of the pole (negative x)

These are the equations you will use in order to find the conic equation with the given information you are given.

In addition, you must be able to identify an ellipse, parabola, and hyperbola for each conic equation. You can determine this through the eccentricity (e).

ellipse: e< 1

parabola: e= 1

hyperbola: e>1

Lesson 10.6

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Today we learned about Polar Coordinates.  Basically, this is another form of graphing other than using the x and y axis.  In particular, this graph uses circles and radians to graph plot points.  Here is some vocabulary commonly used in polar coordinates.

r = directed distance from O to P
Theta = directed angle, counterclockwise from polar axis to segment OP.

In addition, there are coordinate conversions you need to know in order to be able to understand your points.

Polar to Rectangular Conversion
During a Polar to Rectangular Conversion, you will be given points (r, theta) and you will plug them into the corresponding coordinate conversion equation. (x = cos, y = sin)

Why Calculus?

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I've always wondered why certain math subjects are needed for specific occupations.  For instance, Calculus is needed to become a Pediatrician...um what?  I know this because I am very interested in becoming a pediatrician and when I looked at what college courses were necessary, Calculus was one of them!  So I decided to search on what's so cool about Calculus.
Well, to start of you actually don't REALLY need Calculus to be a Pediatrician.  You actually need it to go to Med School.  Calculus is a concept that is found in many sciences, such as Physics and Chemistry, which are subjects that are part of the MCAT (Medical College Admission Test).  Though one can go through the MCAT using Algebra-based concepts, one will not receive an adequate score compared to one who uses Calculus-based concepts.  Getting a good score on the MCAT is important because it is a ticket to get you into a Medical School, and getting into Medical School is how you become a Pediatrician.
All in all Pediatricians don't need Calculus, but you DO need Calculus to become a Pediatrician.

Lesson 10.3

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Today we learned about hyperbolas.  Hyperbolas are a set of all points (x,y) the difference of whose distances from two distinct points (foci) is constant. Hyperbolas consist of two parabolas that are equal, but are positive and negative. The vertices are equidistant from the center.

Here are some equations you need to know.

Your asymptotes are used so that you know the boundaries of your hyperbolas. 

Here's an example:

Here's how to do it!

Step one: Turn the equation into standard form.

Step two: Find your a, b, and center (h,k)

Step three: Find your vertices using a and b.

Step four: Draw a square around your vertices, this will help with your asymptotes.

Step five: Place your asymptotes by drawling lines from the corner of your square through the center.

Step six: Find your foci.

PI DAY!

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Today (well, tomorrow) is Pi Day, March 14, 2014.  Pi, is a Greek letter that represents a constant. It is most frequently used in equations such as the circumference or area of a circle.  Pi has been known for over 4,000 years and nobody really knows who discovered it.  It has so many digits after the decimal that it's impossible to memorize every digit of pi.  Since Pi Day is such a glorious holiday in the Math World, and our small town of MathLand, I found a Scavenger Hunt that is based off of Pi!

Here are the answers I got.

3. cylinder and sphere

1. Savannah, Georgia

4. E,S,I,O

1. Indiana 1897

5. C = 2 pi r (circumference of a circle) A = pi r^2 (area of a circle)

9. target, pepsi, at&t, no smoking

2. Round Rock, Texas and Cannon Ball, North Dakota

6. Colorado, Florida, Idaho, Kansas, Kentucky, Minnesota, Missouri, Montana, and Nebraska

5. Golf, Volleyball, Basketball, Softball, Baseball

3. Albert Einstein 1879

5. Around the World in 80 Days

8. M&M's, Skittles

9. Ring of Fire, Joy to the World, Here Comes the Sun, Circle of Life

7. allrecipes.com/recipes/desserts/pies

Lesson 10.2

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Today we learned about elipses. Elipses are a set of all points (x,y) the sum of whose distances from two points fixed points is constant. The equation of an elipses is determined through its major axis.  The major axis is where the elipses is longer in length.
Here are the equations and terms you need to know for elipses!

Here's an example!

Lesson 10.1

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Today we learned about parabolas.  A parabola is a set of points (x,y) that are equidistant from a fixed line (directrix) and a fixed point not on the line (focus). There are two equations you need to do and they correspond with which way the parabola is facing and opens up.
When a parabola is vertical it is equal to (x-h)^2 = 4p(y-k).
When a parabola is horizontal it is equal to (y-k)^2 = 4p(h-k).
P = the distance between the vertex and the focus = the distance between the vertex and the directrix
Find P with the coefficient of the equation equal to 4p.

Here's an example!

Human Calculator (Not Miss V)

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Even though we all think that Miss V is the Human Calculator, SHE'S NOT! It's actually a man named Scott Flansburg.  He goes around the world to show that calculators aren't everything, that it's really all about his brain.  He does everything that you can use on a calculator, even square roots and fractions!  He even beat a room of mathematicians at SDSU. It's actually so crazy how he is able to recite a list of numbers, it's amazing.  I don't know what he does everyday, or how he is able to calculate such things faster than a calculator, but obviously I'm a little behind on my math compared to him.  It's even said that maybe a total of 6 people IN THE WORLD, which consists of billions of people, have the talent Flansburg does.  You should watch this video and see this unique talent for yourself!

Exploring Data

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Do you remember mean, median, and mode? Well, you should because it's one of the simplest things in math and is actually used very often.  Even in many jobs today.  But let's just review. Mean is the average of a set of numbers, the median is the middle number when they are put in order from least to greatest, and the mode is the number that occurs the most. Finding mean, median, and mode is what central tendency is! Easy peasy. Another thing used with mean, median, and mode is measures of dispersion.  This is when you find variance and standard deviation. First you find the mean, then you plug in the numbers into the variance equation, and then find the square root. And that's how you find standard deviation!

Probability

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This week we learned about probability. Probability is the likelihood of something happening through the situation and information at hand. Probability deals with outcomes, sample space and an event.  The outcome is the result of the experiment, the sample space is the possible outcomes, and the even is the subcollection of the possible outcomes. The probability of an event is described as the event over the sample space. An independent event occurs when order does not matter and a occurrence has no effect on the second occurrence. For instance when a marble is taken, but then put back, the event is not changed.  Probability questions are usually word problems, and they just take a deeper understanding of the problem.  Then, you can use your common sense, or the equations to find your answer.

Probability of a Complement: P(A) = 1 - P(A)

Probability of Independent Events: P(A and B) = P(A) x P(B)

Probability of an Event: P)EP = n(E)/n(S)

The Language of Math

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Sometimes I forget how much math I've learned over the years, including the different vocabulary!  In addition, I realize how funny it can be for people who have never learned some "Math Language" and how they choose to solve a problem.

I find this picture so funny! I feel like if I never learned what "finding x" really meant, this would be me! But, I don't think there's an answer like this on the SAT.

Though that was a little funny, there are really mathematical terms that I have learned that mean a lot.  For instance, when we have to find an answer in terms of z.  I think this is one of the most confusing concepts, because we use variables in order to find an answer without an EXACT answer.  Also, the fact that there is a way to verify and simplify using sin, cosine, and tangent.  Honestly, things get so confusing with all these math terms.  And we're only going to learn more.  Eventually, we will be able to find different real-life solutions using the language of math!

The Binomial Theorem and Pascal's Triangle

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The binomial theorem is the expansion of a binomial to the nth term.  It seems like you would have to multiply and multiply continuously, but there's always a shorter way! Well, at least there is one this time.  For the binomial theorem, in order to find a certain term, you can use combinations, which include factorials.  The equation for such combination is: nCr = n!/(n-r)!r! But, there's something much simpler in order to find a total equation.  Which is using Pascal's Triangle. Pascal's Triangle is a triangle that distributes the coefficients for a binomial.  But, it's not that simple, you must still multiply that term by others as well.

Here are some examples of Pascal's triangle.

Well-Ordering Principle and Mathematical Induction

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The well ordering principle states that every non-empty set of positive integers contains a least element. Meaning that there is almost a smallest term in a set of positive integers. This principle is also equivalent to the well-ordering theorem. This includes a set of numbers, with a well-ordered subset, in which every nonempty subset contains a least element.  The whole set will depend on the framework of the natural numbers as well as the second property of numbers.  These are known as an axiom or a provable theorem.
Here's a video on the process of the well-ordering principle.

Something equivalent to the well-ordering principle is mathematical induction.  Which is what we applied in our lesson today.  Mathematical induction is basically proving how a set of numbers work, and if true, proving how it will work for any variable put in.  There are two steps to mathematical induction, and it is a bit tricky.  But once you get it, it's quite easy.

Q

Personally, the hardest part for me is in step two. Mostly remembering that your previous term is equal to your given, and then substituting (n+1) for n.  In addition, simplifying and verifying that both equations are equal is a bit tricky, but once you remember your factoring, it gets a lot easier!

"Do I need this for my future?"

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Whenever we come across a lesson that seems utterly boring and not fun, I always say, "Am I ever going to use this?" or "Do I need this for my future?" And I think many students think the same thing.  But, apparently MANY students do, which is why a group of mathematicians have created a website called weusemath.org  This is where I found a video on why they believe math is really important.

The video is narrated by a few people who majored in math and ended up in various jobs.  Professors, scientists, analysts, and many other jobs use math within their works.  In addition, the video also talks about how majoring in math is a good basis for any job!  You have various amounts of options to choose from.  But, I realized that these people have a real hard-core passion for math.  One man even said that everyday he is excited to go to work and he has a big smile on his face.  I definitely can see why math can be a good use to many jobs out there, but I don't think I have the passion or love for math to have it as a job. #sorrynotsorry

Lesson 9.2

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Today we learned about arithmetic sequences, which is a sequence whose consecutive terms have a common difference.  An easy way to find out if a sequence is arithmetic is by subtracting the first term from the second term.  This will be the common difference, if the common difference is sustained throughout the whole sequence, then it is an arithmetic sequence.

In order to find the nth term of an arithmetic sequence, all you have to do is keep adding numbers until you get to that term! Just kidding, there's a formula (because there's one for everything!) and it makes things so much easier.
 The formula is An = A1 + (n-1)d
A1 is the first term of the sequence.  N is the number term you want to find. And d is the common difference.
Here's an example!

In addition to arithmetic sequences, there is the sum of arithmetic sequences! And of course, there is a formula!
The formula is Sn = n/2(A1 + An)
N would be the last term you want to find.  Then A1 is the first term and An would be the last term.  But, something that most people get confused with is putting the number of the term in An, when it is supposed to be the actual term.  So you must find the actual term first by using the explicit formula.  Here is an example:

Lesson 9.1

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Today we learned about sequences and summation notation.  This is a bit of a review from Algebra 2 last year, but with a few added steps.  To be able to solve a sequences and summation notation, you must start with the basics!

When given an equation, the variable n stands for the number of the term you are looking for.  All you have to do is plug in the wanted term, and you are able to solve for the term needed.
Here is an example:

The next thing you are able to do with sequences, is find the nth term.  Which is a term that can be found with any n given.  But, sometimes you are not given an equation.  When this occurs, you must first create an equation, and then plug the n in and you will get your term.  In order to find the equation, you must use the terms given to you, as well as the number in the sequence they correspond to.  After, you must use you brain (tough, I know) and find how the equations relate to n.  Once you find your equation, you just go back to your easy ways and plug and chug and there you have your nth term!
Here is an example:

Factorials, factorials, factorials! These were not my favorite in Algebra 2, especially because they are just dumb.  But, surprisingly factorials are sometimes used in sequences as well.  They may be given in an equation!  Just in case you forgot, a factorial (n!) means you must multiply the number given as well as the numbers preceding it. (So you might end up with a big number).  But sometimes, factorials cancel out, which is great!
Here's a simple explicit formula with a factorial.

Finally, summation notation!  It sounds really hard, but it is quite simple once you get it.  Basically a summation notation gives you a range of numbers, which will be your n's, and an explicit formula.  You must find every term needed and then add them up! And that's the answer to your summation notation! It seems a little bit confusing, but once you get it, it's a piece of cake.
Here's an example of a summation notation problem.

Chapter 8

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Chapter 8 was all about matrices and determinants.  We learned different ways to use matrices in order to solve a system of equations, in addition matrices are used to find determinants.  We also found out 4 applications of matrices and determinants.

Here's a link to our website all about Chapter 8!

Four Applications

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Today we learned about four different applications for determinants in matrices.

• Area of a Triangle
• Test for Collinear Points
• Equation of a Line
• Cramer's Rule

Here are explanations and examples of each type of problem.

Here is a voicethread on an example of an area of a triangle.

Determinants of Matrices

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Today we learned about finding determinants of matrices.  The process is simple, but it gets a lot more difficult with bigger matrices.  There are different tricks you can use in order to solve matrices quicker as well.  But basically, all you have to do is multiply diagonally to the right and then subtract the opposite diagonal.

Here are some examples.

Determinants eventually get a lot harder to solve once your matrices get bigger.  Here is a video on how to solve such determinants.

Cryptography

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Cryptography coding is an application in which you use matrices and their immerses to create a code. You start off with the inverse of an encryption matrix, which will be your decryption matrix, also known as your code. Next, you use the numbers given, and multiply by the decryption matrix to find the code corresponding letters of the alphabet to numbers. Here is my cryptography worksheet, which includes me solving Mama V's message, as well as a message I created.

Can you solve my message? Don't forget to correspond the code numbers to the size of your decryption matrix. For mine, you will separate groups of numbers by two!

Lesson 8.2

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8.2 is all about outing a system of equations into matrices and solving. It's quite easily, especially when you use the Gauss-Jordan formula. But sometimes you will encounter a system with no solution as well as a system with an infinite number of solutions.

You can find out a system has no solution when one of the rows ends up with all zeroes. This means that zero will equal a number, which would not be true.

Finding a system with infinite solutions is just like finding an equation in terms of z. When this occurs, you will have an infinite number of solutions.

Lesson 8.3

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Today we learned about inverse matrices and how to find them. Basically we make the given matrices equal to their corresponding diagonal ones, and then the answers are the inverse. It's a lot easier a simpler once you see an example.

But, there are times when there is no inverse. This occurs when the bottom row of the original matrice ends up as all zeroes. That is when it is called singular.

Chapter 8 Vocabulary

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Here's our prezi on the vocabulary for chapter 8.

http://prezi.com/poqksnaiqwz_/?utm_campaign=share&utm_medium=copy

Lesson 8.1

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Today we learned two different methods of elimination using matrices, the Gaussian Elimination and the Gauss Jordan Elimination.  These two methods are somewhat similar, but personally, I think the Gauss Jordan Elimination is way easier and faster.

The Gaussian Elimination deals with inputting a given system of equations in matrices, and then creating your diagonal of 1's, which will give you a new set of equations.  Eventually you will have a given solution to one of your terms, and then you just plug and chug and you'll be able to solve for your other variables.

The Gauss Jordan Elimination is a little different, because it deals with finding your constants within your matrices, just by making all the numbers outside of your diagonal 1's zeroes.  It gets a little tricky to do this, but once you do, everything is really simple and your answers come right on the page.

Here are some examples:

#3

In addition, referring to the error analysis, the problem was incorrect due to the fact that the student did not correctly zero out the first zero, because he or she did not ally the (-1) to the whole row. In conclusion, the answer should actually be (7,-3).

The Unsolvable Math Problem

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As I was looking for things to post today, I came across an article about a college student who had ACCIDENTALLY solved a math problem that mathematicians since Einstein have been trying to solve.
The student had been studying very late the night before a test and ended up sleeping in and attending the class late.  When he arrived, he quickly began the three problems that had been posted on the board. He was having a hard time on the last one, and it almost seemed impossible, but he suddenly found a method that worked and was able to finish just before time was called.  Later that night, the student received a call from his professor whom seemed frantic.  The student thought that he had failed the whole test, but in reality, the professor told the student that the third problem was an example of an "impossible equation" and the student was able to solve it!  The student was quite amazed with himself, especially since he wasn't even supposed to do the problem in the first place!
So I guess you could say that you never what's going to happen when you just do all the problems on the board, maybe you'll find a solution to something that hasn't been solved in centuries!

The name of this man was George Dantzig.  During the time of his revelation, he was a student completing his Doctorate at UC Berkeley in 1946.

Math Soup for the Body and Soul

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Not really, but I always thought the title of those books were funny, which I thought would be a good title for a post about math jokes.

How do you make seven an even number? 

You take the s out!

Why should the number 288 never be mentioned?

It's two gross.

Why did I divide sin by tan?

Just cos.

What do you call friends who love math?

Algebros.

Even though these were a little corny, I still found them pretty funny, but also kind of sad that people could even think of Math jokes. This just shows how everybody has their own preference, and that some people actually enjoy math and wouldn't mind using it as a subject of a joke, or of a talk, or of anything really.

Chapter 7 Review

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Chapter 7 was filled with lots of different concepts including old and new. From elimination am substitution, to linear programming. Another thing we learned that was somewhat new but also old was breakeven. There's a possibility this was done in Algebra 2, but all it involves is substitution, which was definitely from algebra. Break even is used to find the point at which a business' cost and revenue are the same. In order to find this, you must use two equations.

Total revenue= (price per unit)(# of units sold)

Total cost = (cost per unit)(# of units sold) + initial cost

Here's an example:

Sorry this i late, but I'm battling illnesses.

Dyscalculia

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Dyscalculia means difficulty in learning arithmetic, such as difficulty in understanding numbers, and learning math facts!  This is a disability similar to dyslexia.  But, it is less common and only exists between 3 and 6% of the population, in addition a quarter of children with dyscalculia have ADHD.

Dyscalculia comes from Greek and Latin which literally means "counting badly".  This disorder might seem quite silly and an excuse to have a bad Math grade, but it is a real disorder with many problems that come with it.  For instance, one with dyscalculia can have difficult reading analog clocks, difficulty stating which of two numbers is larger, problems with differentiating between left and right, as well as an inability to concentrate on mentally intensive tasks.

Though people with Dyscalculia have a more difficult time with math, there are treatments to help remediate it.  For instance, forms of educational therapy as well as direct stimulation have been proved to demonstrate selective improvement in results.

Personally, I think that Dyscalculia is a disorder that many people are unaware of, and that there's a possibility that I could have it too! Especially when it comes to Math Analysis...

Lesson 7.5

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Systems of inequalities are the next step after solving a system of equations, but we'll still be using everything eventually. In order to solve a system of inequality, a graph will be needed in order to find the solution that will satisfy all the inequalities.

 

Here are the steps:

Replace the inequality sign with a equal sign, and sketch the graph of the resulting equation.

 >dashed lines (greater than or less than) 

 >solid lines (greater than or equal to or less than or equal to)

Test one point in each regions formed by the graph in step 1. If the point satisfies the inequality, shade the entire region to denote that every point in the region satisfies the inequality.

Solution of a system of inequalities in x and y is a point (x,y) that satisfies each inequality in the system

For a system of inequalities it is helpful to find the vertices of the solution region.

Here's an example: 

Just always remember to check which vertices satisfy the inequality that includes x and y!

Lesson 7.4

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We finally left easy peasy Algebra, and now we have to do these wild things called partial fractions. To be honest, they're much scarier at the beginning, but once you do a ton of practice, it's super easy. Especially if you follow the steps!

These steps are:

Multiply by the lowest common denominator

Distribute

Collect terms with the same variable

Factor out the variable

Equate coefficients

Solve the system of equations 

Write as a partial fraction

Here's an example:

It's quit simple, you just always have to remember your keep your terms organized and you'll get it easily.

I feel like Math Analysis is going to get harder, so we should all buckle up and get ready!

Lesson 7.3

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Today, we used elimination, but with three variables, adding another step.  Another thing we learned was a non-square system.  Here's an example of a system of equations with three equations.  

Educreations video by Joe and Axie

Underwater Math

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All week I've been thinking about how on earth I was going to think of a topic for our creative math post.  Some came up, but they were truly boring and didn't actually apply to my personal life really.  Until today, during swim practice, I realized that I use math ALL THE TIME, so do many other sports.  Obviously it's not just the simple adding of goals, touchdowns, and points in team sports, but also math related in time, especially in swimming and track.

For instance, today Coach Simon and Zimbrano taught us about a "Russian Time", which was something completely new to us all.  Basically, it's a time we received on a 50 yard swim, plus an average amount of strokes we took per lap.  The goal at the end of the set, was to reduce your Russian Time, but keep the average stroke count the same.  

Another thing I realized was how swimming is solely based on time standards, which really correlate to math in practice.  For instance, in order to reach a certain time during our meets, we practice by going a certain average time in practice, making sure we hit that pace every single time.  Eventually, our goals get faster and we have to calculate it all again.  In addition, when we're practicing.  We use math all the time, because every swimmer calculates their own personal time using the clock on the scoreboard.  So basically, we all become professionals at the 60 second clock system, being able to understand how much we need to subtract compared to the time we leave.  We also figure out our averages by dividing bigger times, finding the average we need for shorter distances.

Even though the math we use isn't as complicated as parabolas, trig functions, or systems of equations, it's still math and it's all mental math.  Which is a lot harder than you think, especially when your body is tired AND you still have to use your mental juices.  All in all, math is used all the time, even in your daily routines, such as swim practice.  But sometimes, too much math is just too much and you can't handle it!

Lesson 7.2

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Now it's time to add another way to solve a system of equations, ELIMINATION. In order to eliminate a variable in an equation, you should follow these steps.

Obtain coefficients that differ only in sign.

Add equations to eliminate a variable.

Back substitute to solve for second equation.

Check your solution.

Here's an example. 

The process of elimination can also be applied to real life situations, such as one dealing with the speed of an airplane with a headwind compared to the speed of an airplane with a tailwind.

Here's an example.

And that's how you use elimination to solve a system of equations. So far second semester is good! Hopefully we can keep it up! 

Lesson 7.1

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It's the first day of second semester and Ms. Van Spronsen is already giving us a hard time. Today we learned TWO things, substitution AND breakeven. Which was way too much for one day. 

Anyways, substitution is really easy peasy and quite obvious. All you have to do is isolate a variable on one equation then substitute it in the other. Then you solve and then back substitute.

And that's how you substitute! Now, onto breaking even.

There are two equations used to find your break even, which is total cost = total revenue

In order to find your total cost, you multiply your cost per unit and numbers of units sold, then add the initial cost.

In order to find total revenue, you multiply the price per unit and the number of units sold.

And that's how you find the break even. Basically you just need to make sure you're plugging everything in its correct place and just go back to using substitution for the missing variables!

Well that's Day 1 of Second Semester and blogging!