Conic Sections Review

Parabolas:

The basic equations for a parabola are (x-h)^2 = 4p(y-k) and (y-k)^2 = 4p(x-h).

P is the distance between the vertex and its focus, which is in the same direction as the parabola, and between the vertex and the directrix, or a line directly behind the parabola. (h,k) are the rectangular coordinates for the vertex.

Ellipses:

The basic equations for ellipses are:

[(y-k)^2]/a^2 + [(x-h)^2]/b^2              AND          [(x-h)^2]/a^2 + [(y-k)^2]/b^2       (h,k) are the rectangular coordinates of the ellipse's center. a is half of the major axis (longer "diameter") and b is half of the minor axis (shorter "diameter"). Hyperbolas:  Once again, (h,k) are the coordinates of the center. A and b are the distances from the center to the edge of a rectangle through which the asymptotes of the graph pass diagonally. A will also be the distance from the center to the vertices of the hyperbola

Rotating Conics Review

In this section, we learned how to find the equation and sketch the graph of a conic section that is not perfectly vertical or horizontal, but on tilted axes. This picture shows the three basic equations needed in this chapter:


Steps:

First, you must use the top equation to find the angle of rotation of the axes. You then use that angle to simplify the bottom two equations. The ultimate goal is to replace x and y in the original conic's equation with x' and y', or the new x and y that correspond with the new axes.

Parametric Equations

Parametric equations involve a third variable, usually t. This allows us to depict the passage of time on a graph. The first step is to eliminate the parameter, usually by substituting to equate the x and y variables. Then, graph the regular rectangular equation on a graph. Finally, insert a few values for t in the original equations and label them on the graph. Below are two example problems:

Summations and Limits to Find Area

The following picture shows the six equations used in this lesson:

Here is an example of a problem that utilizes nearly all of the above equations:

Equation #6 is used throughout the first half of this problem. The main strategy is to factor 1/n out of the equation, then putting it outside of the summation. Once 1/n is removed three times so it's 1/n^3, you have to split up the summation into three summations using rule #5. You then use rule #1, #2, and #3, respectively, to simplify each summation. Simplify, then take the limit.

Determinant of a Matrix

There are several strategies to finding the determinant of the matrix. The basic strategy for a 2X2 matrix is to multply the top left number by the bottom right number and subtract the product of the two remaining numbers. Anything bigger than a 3X3 requires a complicated process of finding minors and cofactors that can be seen in action here:

http://www.educreations.com/lesson/view/determinant-of-a-matrix/16850106/?s=ciem9G&ref=link

For 3X3's, you can rewrite the first two columns of the matrix to the right of the matrix and multiply numbers diagonally, then add and subtract. This strategy is also seen in that Educreations video.

Verifying Trigonometric Identities

This section features many problems in which a combination of trig functions must be changed to look like another set of trig functions. These problems are not that difficult now, as we have memorized many trig identities. Strategies on these problems:

• Work with only one side of the equation (choose the more complicated side). 
• Try to convert everything to sines and cosines.
• Simplify complex parts of the problem ASAP using trig identities.

NBA Real Plus-Minus

ESPN just came out with a new advanced metric called Real Plus-Minus (RPM). It is intended to be a perfected form of the normal plus-minus stat, which measures the gains or losses of points while a given player is on the field. The issue with the original plus-minus stat is that it doesn't factor in the impact of the other players on the court at the same time as the given player, so mediocre players who play alongside superstars will often excel in this stat as much as their superior teammates.

Nothing has been revealed about the actual calculation of this stat yet, but it allegedly factors in numerous variables on both offense and defense to calculate the number of points a player contributes to or takes away from his team per 100 possessions. Although it's difficult to do anything to calculate using this metric because the algorithm is not given, we can use the already-calculated RPM numbers to compare and contrast different teams.

Comparison of the Oklahoma City Thunder and the Miami Heat based on the RPM of starters:

OKC:

Russell Westbrook: 4.09

Thabo Sefolosha: 0.37

Kevin Durant: 6.77

Serge Ibaka: 3.10

Kendrick Perkins: -3.19

Total: 11.14

MIA:

Mario Chalmers: 1.48

Dwyane Wade: 2.13

Shane Battier: 1.60

LeBron James: 8.11

Chris Bosh: 3.75

Total: 17.07

Interestingly, Nick Collison, a backup big-man for the Thunder, is the sixth-best player in the NBA according to this statistic with an RPM of 5.81. Collison is the same height as Kendrick Perkins, whose RPM is a lowly -3.19; if they inserted Collison into the starting lineup for Kendrick Perkins, their RPM total for starters would be increased to 20.14! 

Although this stat is new and certainly imperfect to some degree, it does bring the value of some underrated players to light. Maybe this switch would improve OKC and make them even better than the defending-champ Heat.

Source:
http://espn.go.com/nba/statistics/rpm/_/sort/RPM

11.2 Vectors In Space

This section is about the properties of vectors in 3-D space. The component (regular) form of a vector is
v = <v1,v2,v3>. As we know from earlier chapters, a vector is the difference between a terminal and initial point. Since this is in three dimensions, the three numbers inside the angle brackets are the differences between the x-values, y-values, and z-values, respectively, of a terminal and initial point.


Length of a vector: ||v|| = squareroot(v1^2 + v2^2 + v3^2)

• This equation makes sense, as the length of a vector would be the distance between its initial point and its terminal point. Each vn value is the difference between the x-, y-, or z-value of the terminal and initial points, so this length equation is the same as the basic distance formula we all know and love!

11.4 Lines and Planes In Space

In this lesson, we learned how to find the parametric and symmetric equations of a line in a 3-D space and how to find the angle and distance between two planes. In this blog, I'm going to focus on planes. The basic equation is:

a(x-x1) + b(y-y1) + b(z-z1) = 0

In this equation:

• a,b,c are the x, y, and z components of a perpendicular (normal) vector. n = <a,b,c>
• x1, y1, and z1 are the coordinates of a point on the plane.
• x,y, and z are variables. 

Many problems will give hints and ask you to find the equation of a plane. In the following example, a point on the plane and a perpendicular vector are given:

Cool 3D Shapes

The Importance of Steals

I read an interesting article about the importance of steals in the NBA last night. The writer, Benjamin Morris, measured the performance of teams with and without certain players. He compared the team's points per game with and without the player to the player's average stats and a replacement player's average stats; the chart below shows his results. For example, if a team is playing without a player that gets 12 rebounds per game who is being replaced by a player with just 2 rebounds per game, the team will probably average 17 fewer points.

As you can see in this chart, a steal is 9 times more valuable than a point! It's high value is understandable, as a steal will almost always result in two easy points in the resulting fast-break scoring opportunity and also shifts the momentum, but it's amazing that such an overlooked stat is so valuable. It's easier to glorify high-scoring players because the numbers are so much grander; Kevin Durant is leading the league with 32.2 points per game, while Chris Paul leads the league in steals with 2.53 steals per game. But let's look at the stats of these two players with the logic of this chart:

Durant:

32.2 PPG

1.3 STLPG

0.8 BLKPG

5.6 APG

7.6 RPG

3.6 TOPG

32.2(1) + 1.3(9.1) + 0.8(6.1) + 5.6(2.2) + 7.6(1.7) - 3.6(5.4) = 54.71

Chris Paul:

18.5 PPG

2.5 STLPG

0.1 BLKPG

11.0 APG

4.3 RPG

2.4 TOPG

18.5(1) + 2.5(9.1) + 0.1(6.1) + 11(2.2) + 4.3(1.7) - 2.4(5.4) = 60.41

With the logic of a player's contributions compared to a replacement player, it seems like our own Chris Paul of the Los Angeles Clippers is an even better player than Kevin Durant, the leading MVP candidate!

Works Cited:

http://espn.go.com/nba/player/_/id/2779/chris-paul

http://espn.go.com/nba/player/_/id/3202/kevin-durant

http://fivethirtyeight.com/features/the-hidden-value-of-the-nba-steal/

10.8 Polar Equations of Conics

In this section, we are finding the polar equations of conics and graphing them on polar graphs. In this example, we are given the type of conic (parabola), the eccentricity (1), and the directrix (y=-2). With a horizontal directrix below the origin, we can tell that the equation's denominator will be (1-esinØ). The directrix is also 2 units away from the focus [which is always at (0,0) for these problems], so we know that p is 2. The bottom row of this picture displays how to insert those numbers and find the final equation (boxed on the right).

10.6 Polar Coordinates

This section introduces us to graphing using polar coordinates rather than the usual rectangular (x,y) coordinates. Polar coordinates are (r,Ø) [that Ø is the closest thing to theta I could get]. The four equations in the picture below are used to transfer polar equations into rectangular equations and vice versa. 

This next picture is an example of changing a polar equation into a rectangular one. Knowing the angle, you can plug it into the top-left equation from the above picture. The final answer is boxed below:

This printable paper is helpful to use for graphing polar equations:

http://www.embeddedmath.com/downloads/files/polargraph/polargraph-letter.pdf

Wilt Chamberlain vs. The Modern NBA

In my free time, I like to look at stats of NBA legends to understand why they were so famous and successful. Today, I was reading about Wilt Chamberlain and I saw that in his 1961-1962 season on the Philadelphia Warriors, he had an average of 48.5 minutes per game in a game where there are only 48 minutes per game. I researched the history of the length of NBA games to see if they were longer back then or something, but I found that they've been 48 minutes since the beginning of the NBA. The only reason Wilt averaged more than 48 minutes per game is because of the few games that went into overtime.

On basketball-reference.com, I found that 10 overtime periods were played by the Warriors that season (5 single-overtime games, 1 double-overtime game, and 1 triple-overtime game). Each overtime period is 5 minutes long, so since there were 10 OTs, 50 extra minutes were played.

There were only 80 games in the season at the time, so the number of regulation minutes that season (48X80) was 3,840. With the 50 OT minutes, the total was 3890. Wilt played 3882 minutes that season. 3882/3890 = 99.79% of the total minutes played. Interestingly, I found on Wikipedia that the only reason he didn't play those 8 minutes was that he was ejected from one game with 8 minutes left after getting his second technical foul.

His 3882 minutes over 80 games in a season (3882/80) means he averaged exactly 48.525 minutes per game.

Wilt Chamberlain was 25 years old that season. To compare to a modern NBA superstar, Kevin Durant is 25 years old this season and averages 38.4 minutes per game this season. This number is already ten less than Wilt's average at the same age, and is probably higher than it would have been because Oklahoma City's other superstar Russell Westbrook missed 31 games this season. But even at this pace (in today's 82-game season), Durant would play 3148.8 minutes, over 700 minutes fewer than Chamberlain! This is certainly partially why Chamberlain retired after 14 seasons, while current NBA stars can last much longer. For example, Tim Duncan is in his seventeenth season on the San Antonio Spurs and still plays at an elite level. Because of increasing awareness about injuries and the fact that athletes usually want their career to last as long as possible, I don't think we'll ever see a player play as many minutes as Wilt again.


Works Cited:
http://www.basketball-reference.com/teams/PHW/1962_games.html
http://en.wikipedia.org/wiki/Wilt_Chamberlain
http://espn.go.com/nba/player/_/id/3202/kevin-durant

10.4 Rotation of Conics

In this section, we learned how to find the equation and sketch the graph of a conic section that is not perfectly vertical or horizontal, but on tilted axes. The following picture is of three equations that are necessary to start every one of these problems:

First, you must use the top equation to find the angle of rotation of the axes. You then use that angle to simplify the bottom two equations. The ultimate goal is to replace x and y in the original conic's equation with x' and y', or the new x and y that correspond with the new axes. The next picture shows the classifications of each type of conic section using its determinant. From the value of the determinant, we can tell which type of conic section an equation represents before completing the problem. This will be helpful to double-check the result.

10.3 Hyperbolas

Here are the equations for a hyperbola:

The first equation is used for horizontal hyperbolas, while the second is used for vertical hyperbolas.

Below is an example of a problem that asks you to sketch a hyperbola. Unless the coefficients of x^2 and y^2 are already one, the first step to these types of problems is completing the square. This is done on the second line of this problem. (Don't forget to add equal amounts to the other side of the equation!) to sketch the graph, mark the center (h,k), then mark points that are "a" distance away in the direction "a" corresponds to (in this case, x). Do the same for "b" (in this case, in the y direction) and make a box; the asymptotes, or lines the graph will never touch but approach infinitely, will run diagonally out of the corners of the box. Then draw the graph as shown, leaving from the vertex, or the "a" points you marked, and approaching the asymptotes.

10.2 Ellipses

This section focuses on graphing and finding equations for ellipses. The basic equations for ellipses are:

[(y-k)^2]/a^2 + [(x-h)^2]/b^2              AND          [(x-h)^2]/a^2 + [(y-k)^2]/b^2            (h,k) are the coordinates of the center. "a" and "b" are the distance between vertices and the center; a is the longer of the two. Although "c" is not in the equation, it is important to know that c is the distance between the center and each focus. c2 = a2 – b2 Some problems will give you the equation of an ellipse and you have to find out information from it (center, foci, vertices, and eccentricity) and sketch it using that information. Other problems will give you hints and ask you to find the equation. This problem (#33) gives you a sketch and the vertices, and asks for the equation:

Works Cited:
http://www.purplemath.com/modules/ellipse.htm

The REAL True Shooting Percentage

So there's this stat that ESPN NBA analysts use all the time called true shooting percentage. It is intended to measure a player's shooting ability more accurately, as it accounts for free throws and three-pointers. However, when I saw the equation used to calculate it, I didn't understand how it accomplished its goal at all. Here's the equation:

True Shooting Percentage = Total points / [(FGA + (0.44 x FTA)]

In my research, I learned that the equation was created to account for the number of possessions a shot consumes. This helps explain the (.44 x FTA) part of the equation, as usually, the other team gets the ball after a pair of free throws, and some free throws come after a made shot on a three-point play, so such a free throw is a chance for points that doesn't even take up a possession. 0.44 was reasonable because it was slightly less than one-half, or .5, and therefore accounted for the possessions a free throw consumes fairly well. 

However, this means that "True Shooting Percentage" is a misleading name for this statistic; it is more a measure of efficiency per possession than per shot. So, I made my own, more reasonable True Shooting Percentage equation:

Laski True Shooting Percentage = Total Points / [FTA + (2 x FGA) + 3PA]

Or, more simply: LTS% = Points Scored / Total Possible Points 

This equation encompasses all types of shots accurately, as each three-point shot is worth three points,each regular field goal is worth two, and each free throw is worth one. The resulting percentage represents the percentage of points a player scored out of the possible number of points they could have scored if they never missed a shot of any kind.

Stephen Curry's Laski True Shooting Percentage so far this season:

1459/[276 + (2 x 1108) + 492] = 0.4889 --> 48.9%

Works Cited:

http://theoldnorthking.blogspot.com/2013/01/true-shooting-percentage.html

http://espn.go.com/nba/statistics/player/_/stat/scoring

10.1 Parabolas

The basic equations for a parabola are (x-h)^2 = 4p(y-k) and (y-k)^2 = 4p(x-h). As you can see, these two equations are very similar, but the variables are just rearranged. The first equation is used to represent parabolas that go up-and-down, while the latter represents parabolas that go side-to-side. The p is equal to the distance between the vertex of the parabola and it's focus or it's directrix. (The focus is a point in the same direction of the parabola, and the directrix is a line that the parabola goes away from). Finally, (h,k) is the vertex pf the parabola.

Passer Rating: Russell Wilson

These are the steps to finding Passer Rating. You must add together the answers of four equations, then multiply by 100 and divide by 6.

Total 1:

1. Divide a quarterback's completed passes by pass attempts.
2. Subtract 0.3.
3. Divide by 0.2 and record the total. The sum cannot be greater than 2.375 or less than zero.

Total 2:

1. Divide passing yards by pass attempts.
2. Subtract 3.
3. Divide by 4 and record the total. The sum cannot be greater than 2.375 or less than zero.

Total 3:

1. Divide touchdown passes by pass attempts.
2. Divide by 0.05 and record the total. The sum cannot be greater than 2.375 or less than zero.

Total 4:

1. Divide interceptions by pass attempts.
2. Subtract that number from 0.095.
3. Divide that product by 0.04 and record the total. The sum cannot be greater than 2.375 or less than zero.

Final Steps:

1. Add the four totals you recorded.
2. Multiply that total by 100.
3. Divide by 6.
4. The final number is the passer rating.

I used this to calculate Russell Wilson's Quarterback Rating in the Super Bowl:

(2.1+1.31+1.6+2.375)X(100/6)=123.08

Works Cited:

http://football.about.com/od/frequentlyaskedquestions/ht/How-To-Calculate-A-Quarterback-Rating.htm

9.8 and 9.9 - Exploring Data

These sections expand on the main ideas of mean, median, and mode that we've known for a long time. But now, Variance and Standard Deviation are added. The equations for these are given in the chapter, but most of the problems involve too many numbers to do in a reasonable amount of time, so it's a good decision to just use standard deviation and variance calculators online. The following example finds the mean, median, and mode of the set 3, 7, 14, 15, 21, 21.

Mode: 21

Median: 14.5

Mean: 13.5

9.7 Probability

Probability problems can seem really complicated, but are often a lot simpler than they appear. At the base, every problem is asking you the ratio of favorable outcomes to total circumstances 

(# of successes/# of possibilities). For example, if you want to get a coin flip to be heads, there is one successful outcome, but two total outcomes (heads and tails), so the probability of getting heads is 1/2. There are twelve face cards in a deck of 52 cards, so the probability of drawing a face card is 12/52 (3/13). Let's try an example:

14.) In a 52-card deck, one card is drawn. What are the chances the number is six or lower?

• Ace, 2, 3, 4, 5, and 6 are six or lower, and there are four of each of those, so 6X4=24.
• There are 52 total cards, so the probability is 24/52 (Simplified: 6/13).

March Madness Odds

Warren Buffett and the owner of the Cleveland Cavaliers are offering one billion dollars to anyone who gets a March Madness (college basketball) bracket perfectly right. This might seem like an outrageously high number, but interestingly enough, they may not even have to pay it! Let's calculate the odds.

1. There are 68 teams that make it into the tournament, and since each team loses once except the winner, there are total games.

2. Let's assume that you have about a 50% chance of guessing each game correctly. [This might even be an understatement, because some really good teams will play really bad teams (especially early on) and will have a more than 50% chance of winning.]

3. After each successful guess, your next guess will also be a 50% chance, so your chance of winning the whole thing is 0.5 x 0.5 x 0.5 x 0.5 ... until you get 67 0.5's, so (0.5)^67.

4. (0.5)^67 = 6.78 x 10^-21, or 0.00000000000000000000678. That's about a one in six sextillion!

Now, most experts think that it is possible to guess the outcome of some games with such certainty that the average percentage of guessing a game right could be closer to about 75%. With this calculation, your odds are still about one in four billion. Buffett is only allowing 10 million entrants, so in all likelihood he won't even have to give the check to anyone!

Works Cited:
http://parade.condenast.com/255536/erinhill/hes-gone-mad-warren-buffett-offers-1-billion-for-perfect-march-madness-bracket/

Binomial Expansion

Binomial expansion is basically the opposite of factoring. It takes a basic, fully factored combination of variables and numbers and expands it from a basic form (ax +by)^n. The following picture shows a binomial expansion of the equation (2x - 3y)^4:

To find out the coefficients of each term, use Pascal's Triangle (shown on top). Then, take the first term (ax) and with the first coefficient, set it to the 4th power, then the 3rd power, etc. down to the 0th power. Do the opposite with the 2nd term (by), starting at 0th and then going up to the 4th power. Finally, multiply the terms to simplify. Final answer is in green.

Well-Ordering Principle

According to Wikipedia, the well-ordering principle states that every set of positive integers must contain a lowest or least element. This is a very key element in validating mathematical induction as a way to prove formulas.

IMPORTANT: The goal of mathematical induction is to prove that for any number n, a given equation would find the sum of a given sequence.

The first step in mathematical induction is to prove the equation works if n=1. This is possible because of the well-ordering principle, because it proves that n=1 is possible by saying that every positive-integer set has a first number. The next step is to test for n+1, as since we know the equation applies for n=1, this would prove it applies to n=2. But in that case, since n can equal 2, it proves it for n=3, then n=4, and so on and so forth, proving the equation to be true regardless of the value of n.

Works Cited:

http://en.wikipedia.org/wiki/Well-ordering_principle

Comparison of NBA MVP Candidates

Four of the MVP favorites in the NBA this season are Paul George, LeBron James, Kevin Durant, and Blake Griffin. One simple way to test a player's offensive ability is to measure their points per game:

1. Kevin Durant: 31.5 PPG
2. LeBron James: 26.8 PPG
3. Blake Griffin: 24.4 PPG
4. Paul George: 22.5 PPG

But this statistic will not factor in how often a player shoots the ball. If a player shoots 30 shots per game, it is a lot easier to score 30 points. In this case, a player would only have to make 50% of their baskets to score 30 points just off of two-point field goals, not even taking into account 3-pointers or free throws.

1. LeBron James: 57.4% FG
2. Blake Griffin: 54.0% FG
3. Kevin Durant: 51.0% FG
4. Paul George: 43.9% FG

Now, these statistics clearly only factor in offense. It is difficult to factor in defense with statistics, as the only concrete statistics in this category are blocks and steals. In my opinion, the best stat would be a measure of the field goal percentage of the opposing player whom a player is covering. Unfortunately, this statistic is unavailable on ESPN.com, but it is generally agreed that this is the order of these four players from best defender to worst defender.

1. Paul George
2. LeBron James
3. Kevin Durant
4. Blake Griffin

Since all these players excel in all statistics, it is difficult to use statistics to separate one from another. George is the best defender, but the worst scorer of the four. Most analysts agree that LeBron James and Kevin Durant are the only players with a shot at winning MVP, but even though Durant holds a clear edge in points per game, James' edge in field goal percentage is far more of a deciding factor, because it shows that he knows what is best for his team in every situation and only shoots high-percentage shots. James also has an advantage when it comes to intangibles, because he has proven that he has the talent and leadership to win in the playoffs. Therefore, despite Durant's astronomical scoring numbers that cannot be overlooked, I think James should win his 5th MVP in 6 years.

9.2 Arithmetic Sequences

This chapter focused on arithmetic sequences and how to find the sum of a sequence. This problem combines finding the arithmetic equation for a series and finding the sum of said series. In this problem, we are trying to find the sum of a set of 25 numbers. First, we must find the common difference to complete the equation for the arithmetic sequence equation shown below. Then, we can find a25 using that equation. Finally, we plug in a1, a25, and n (which is 25) into the sum of an arithmetic sequence equation shown at the bottom, and we get 1850 as the answer!

9.1 Sequences and Summation Notation

A few of the things we learned to do in this lesson are:
1. Find terms of a sequence.
2. Write equations of sequences.
3. Evaluate factorials.

1. Sequences are often represented as an equation like an = 2n - 1. To find a specific term, like the third term (a3), plug the number of the term into the equation as n. 2(3) - 1 = 5.

2. Finding the equation of a sequence can be tricky, but the best method is to just list the sequence with the number of each term under it., like so:                                  sequence:     1  3  5  7
                                                                                                                n:     1  2  3  4
The pattern can be determined from there.

3. A factorial (!) is a product of consecutive natural numbers, leading up to the number by which the product is represented.
For example, 5! = 5x4x3x2x1.

A lot of times equations that involve factorials can appear overwhelming at first, but simplification is possible if there is division involved:

Example:   100!/99! -----> The denominator cancels out the first 99 factors of the product 100!, so the only number left is 100. Therefore, the answer is 100.

Most of these problems involve a lot of logical thinking rather than plug-and-chug equations. This is good for some students who understand that logic, but it can be a struggle for other students.

Link to the Google Site for Chapter 8 Review

https://sites.google.com/site/chapter8reviewgroup3/

Educreations: Determinant of a Matrix

http://www.educreations.com/lesson/view/determinant-of-a-matrix/

VoiceThread: Testing Collinear Points

Cryptography

On Wednesday, we learned about a form of cryptography, or code language, that uses matrices. In the first step of this worksheet, we had to use an encryption matrix and code Miss V gave us to find out the meaning of the code. We had to use the inverse of the encryption matrix, which is its decryption matrix, to decode the message into new numbers (3 numbers at a time to multiply by the 3X3 decryption matrix). Finally, we had to transfer those new numbers into letters (A=1, B=2, etc.), and the code spelled out the words "MEET ME MONDAY". I then created an encryption matrix of my own in step 2 and made a code. Can you figure out what the code means?

Recognizing Type of Solutions to Systems of Equations

Solving systems of equations using matrices is very helpful, as there are several tricks to be utilized to simplify the system quickly. The most common use of the Elementary Row Operations are to simplify equations into row echelon form.

Row Echelon Form Requirements:
1. All rows consisting entirely of zeros occur at the bottom of the matrix
2. For each row that does not consist entirely of zeros, the first nonzero entry is a 1.
3. For two successive (nonzero) rows, the leading 1 in the higher row is farther to the left than the leading 1 in the lower row.

From row echelon form, we can find the type of solutions the system will have. In the following example, we can tell that the system will have infinitely many solutions from the row echelon form.

(b) 

                                        

                                  

 Since the bottom row would transfer into the equation 0 = 0, the other equation x + (5/2)y = -1/2 is the only relevant equation. Therefore, any number x will result in a number for y, so there are infinitely many solutions.

Here's another example:

            

                                                        

                                     

                              

If you switch the final two equations at the end of this example, it is in row echelon form. That final line would transfer into the equation 0+0+0=12. Obviously, this equation is incorrect, so the system as a whole is inconsistent and therefore has no solutions.

Sources:
http://tutorial.math.lamar.edu/Classes/Alg/AugmentedMatrixII.aspx