College Prep

Points and Lines

amanda chen March 19, 2010

Given the coordinates of two points, you can find out many things such as

The slope or gradient (these two words mean the same thing) of the line connecting the two points
The equation of the line connecting the two points
Correspondingly, the x and y-intercepts of said line
The slope of another line that is perpendicular to the first line.

Slope of a line

Suppose we have two points with coordinates (3,8 ) and (-2, 2). If there is a line connecting these two points, how do we find the slope of this line?

Subtract the y-coordinate of the second point from the first point: 8 – 2 in this case. i.e. 6
Subtract the x-coordinate of the second point from the first point: 3 – (-2) in this case i.e. 5
Divide your answer in step 1 by your answer in step 2: 6 / 5 = 1.2

In general, if you have two points with coordinates (x₁, y₁) and (x₂, y₂) then the formula for the slope is: y₁- y_2/x_{1 –}x₂

Equation of a line

The general formula for a line is y = mx + c where m is the slope of the line and c is the y-intercept (where the line cuts the y-axis). We have already learned from the previous part how to find the slope of a line given 2 points on the line. Here’s how we find the equation of the line.

Take a point with coordinates x and y e.g. the point (3,8 )) from the previous example
Take y – y-coordinate i.e. y – 8
Take x – x-coordinate i.e. x – 3
Multiply Step 3 by the slope, which we have found to be 1.2 earlier. We get 1.2 (x – 3)
Equate Step 2 to Step 4. i.e. y – 8 = 1.2 (x – 3)
The equation in Step 5 is the equation of the line. Simplify it to get y = 1.2 x + 4.4

In general, to get the equation of a line from two points, find the slope first (call it m) then plug in the values of one point into the following equation to get the equation of the line

y – y_{1 – m(}x –x1) where m is given by the formula for slope: y₁- y_2/x_{1 –}x₂

Intercepts

We can directly read off what the y-intercept is from the equation of the line. From the previous example, we know that the line intersects the y-axis at (0, 4.4). To find the x-intercept, we substitute y=0 into the equation. We then get 0 = 1.2x + 4.4 Rearranging that quickly, we realize that x = -4.4 / 1.2 which is approximately -3.67. So the line intersects the x-axis at (0, -3.67)

Slope of perpendicular lines

In general, if we have a line L₁ and another line L₂ with slopes m₁and m₂ then the two lines are related by the following formula

(m₁) (m₂) = -1

So for the line we found in the example above, the slope of any line perpendicular to it is -1/1.2 which is -5/6 or approximately 0.83

Note that there are infinite number of lines that are perpendicular to the original line example y=1.2x + 4.4 If you are given an addition point that lines on the new perpendicular line, you will be able to find the exact equation of the perpendicular line.

Check out Grockit for SAT and ACT practice!

Tackling the Two Short Reading Passages on the SAT

jonathan peters March 17, 2010

The two short passages are a new addition to the SAT; they are further meant to test your reading comprehension skills. While they are a little less intense than the longer passages, they should not be taken lightly. This is the part of the exam where you can pick up points, but be careful, being methodical counts. Here are some strategies to ace these questions.

1. Read both passages (and find the main idea). This should always be your first step. They are only a paragraph or so each—there is no reason not to. Afterwards you will feel much more prepared for the next couple of questions. For each passage, you should pull out one sentence that explains the author’s main idea. Trips to London are exciting. April is the cruelest month. Whatever it is, make sure you find it. This will help you immensely when you answer the questions.

The passages will address both sides of one issue; usually one will be positive, another negative. Sometimes the topics are particularly boring. My advice: make it relevant. If the topic is computer technology, think of the sweet new laptop your parents are planning to get you once you ace the SAT. Do not be deterred by the subject matter.

2. Be an active reader. Take notes. As you read, make sure you are writing down important details and underlining key phrases. Find the attitude and tone of each author. Is the first passage dismissive, supportive, suspicious, depressed, excited? Write that down and then move onto the second passage and do the same. There will be a question about that.

3. Find the relationship between the two passages. There will almost always be a question “How would the author of passage 1 respond to the author of passage 2?” Prepare for that question. Figure out the passages’ differences in opinion and then use that information to make an informed choice. Does the first author like Mozart’s music? Does the second author dismiss it as derivative? Compare and contrast. Remember, find supporting details. If the second author is not a fan of Mozart, find the sentence where he or she explicitly states that.

4. Read the questions. There are six questions, two for each passage and two comparative ones at the end. Some prefer to read the questions before reading the passages so they will know what to look for. This is a fine way to approach the questions, but remember, there are things you will not understand until you read the passage. If you read the passages beforehand, you will go to the questions prepared to make a choice, however tentative. You can start whittling away answer choices and finding the right answers. This may even save you time in the long run.

5. Find the important details. If the question asks “What does the author mean by the word litigious on line 37?” go to line 37 and look at the word in context. Read around line 37, at least two lines up and down. Always go to the passage to answer the question. Do not even rely on your memory of a few moments ago. It is always good to double-check with the actual words of the passage, which brings me to my final point.

6. Double-check. When approaching the final two comparative questions, you may want to reread the passages once more. These are often the hardest questions in the section; finding the relationship involves reading and understanding the passages. Find key examples where the authors state their main idea. If you already have notes on each passage you will be in good stead.

Please visit the Grockit forum or leave a comment here to discuss further.

Handling Stress in College

jordan schonig March 15, 2010

http://www.flickr.com/photos/saranv/ / CC BY 2.0

Imagine: you just woke up to your roommate’s 6:00 a.m. trumpet practice, losing precious sleep; you have to attend your engineering lecture at 8:00 a.m. and are somehow expecting a pop quiz; your suitemate’s pounding techno music makes it nearly impossible to study for said quiz; after class, you must attend a dorm hall meeting, not mandatory but highly encouraged; your fraternity meeting comes right after the dorm meeting, but it’s all the way across campus–you’ll have to run; oh yeah, did I mention you have a ten-page research paper due the next day?

College stress isn’t always that bad, but at times, your responsibilities and commitments will pile up, and it might seem like there aren’t enough hours in the day. The expected stress that accompanies accumulated responsibilities will not only negatively affect your performance in school, but it can damage your health. Here are a few tips to handle the inevitable stress you’ll face in college:

Sleep: During the first week of dorm life, you will not be on your regular high school sleep schedule. But with so many things to do and people to meet, you’ll be okay with that. After the novelty wears off, though, try to maintain a regular sleep schedule. If you’ve been getting five hours of sleep the last three nights and must choose between a good night’s sleep and a fun night’s partying, make the sensible choice. Your body will thank you. And remember, naps aren’t just for infants and old people; in college, naps are awesome–savor them.
Find Me Time: You’ll soon realize that there is no “I” in “college.” No, I don’t mean that you need teamwork to succeed; I mean you will be so often surrounded by people that you may forget the importance of tranquil solitude. Try to figure out when your roommate is in class so you can schedule that much needed moment of Zen. If that’s impossible, then just take a walk by yourself. Simply escaping the congested den of bodies will do wonders for your stress.
Eat: With so many things to do, you may forget the essentials, like, umm, sustenance? It sounds silly, but many students are so wrapped up in their responsibilities that they forget to eat. Don’t just ignore that grumbling coming from your midsection–it’s telling you something important. And please, eat something reasonably good for you, like fruits, veggies, proteins, etc. Good food will make you feel better, physically and mentally.
Take Breaks: If you’ve been working on your research paper for two days straight, you will likely lose your grip on reality. Just stop. The more you study/work without a break, the less productive you become. Do something you enjoy for an hour or two. Play the guitar. Sing a song. Beat level three on Call of Duty. Better yet, hang out with friends to separate your mind from work. When you return to that paper, you’ll feel refreshed and recharged.
Exercise: Exercise will often be the last thing on your mind when you have many things to do. Don’t feel that you have to exhaust yourself at the gym or run five miles. Even thirty minutes of solid exercise will make a dramatic improvement in how you feel. Exercise releases a spurt of serotonin into your brain, thereby boosting your mood. When responsibilities are clouding your head with anxiety, you can’t afford not to get that boost.
Reevaluate: College will be stressful sometimes–there is no getting around that. Make sure, though, that you are not piling on more responsibilities than you can handle. Somewhere between the extra course units, fraternities, clubs, sports practice, honors classes, and research programs, you may realize that there really aren’t enough hours in the day. If you take on too many things, your performance and your health will suffer. For the sake of your wellbeing, reassess your workload and make the necessary changes.

Faulty Comparisons

amanda chen March 12, 2010

Arthur Miller’s plays tend to discuss more serious subjects than Neil Simon.

Compared to his contemporaries, Picasso’s art demonstrates the cubist ideals more effectively.

Ben and Jerry’s rocky road ice cream is infinitely better than Haagen-Dazs.

Can you spot the faulty comparison’s in the above three sentences? If you can’t, don’t worry, you’ll be a pro at them in no time. The trick to recognizing faulty comparisons is identifying the subject and knowing what is being compared. The rule to then apply is that you must always compare apples to apples and oranges to oranges, meaning that you need to compare like objects. In the first sentence, Arthur Miller’s plays are the subject – note, the plays and not Arthur Miller – but it is being compared to Neil Simon. The sentence is faulty because plays are being compared to a person.

The correct sentence should read:

Arthur Miller’s plays tend to discuss more serious subjects than Neil Simon’s.

or Arthur Miller’s plays tend to discuss more serious subjects that Neil Simon’s plays.

Note the difference an apostrophe makes. By saying “Neil Simon’s”, it implies it is talking about Neil Simon’s plays, not Neil Simon himself.

The second and third sentences demonstrate the same problem. Can art be more effective than the painter’s contemporaries and can ice cream be better than a store? Clearly not, and the corrected sentences ought to read

Compared to his contemporaries’ work, Picasso’s art demonstrates the cubist ideals more effectively.

Ben and Jerry’s rock road ice cream is infinitely better than Haagen-Dazs’.

As long as you can identify what is being compared, you should have no trouble recognizing this mistake. However, the SAT writers can make it tricky for you with statements such as this:

The weather in San Diego is nicer and sunnier than the weather any city in California.

This sentence might appear to be correct: the weather in San Diego is being compared to the weather in Californian cities. However, there is a slight paradox. The sentence states “any city in California” and “any city” also includes San Diego. The sentence is thus comparing the weather in San Diego and the weather in all Californian cities, but San Diego’s weather cannot be better than itself, right? So to be perfectly correct, the sentence should read

The weather in San Diego is nice and sunnier than the weather in any other city in California.

The last type of faulty comparison involves superlatives and comparatives. Superlatives are words like most, best, fastest, strongest that compare 3 or more things. The corresponding comparatives, more, better, faster, stronger are used when comparing 2 things only.

For example, Between Jackie and Michelle, Jackie is the strongest swimmer is wrong because the sentence is only comparing the swimming capabilities of two girls. It should read, “Jackie is the stronger swimmer”

This type of comparison is usually easier to pick out than recognizing the unlike items are being compared. One comparative-superlative pair that you might not know is “between” and “among”. Between is always used to compare two objects, while among indicates that there are three objects or more. For example, between taking a SAT prep class or studying on Grockit, I find Grockit more effective. Whereas, Among the myriad of SAT prep courses available, Grockit is by far the best.

Comparison rules are not hard to bear in mind, but to make sure the lesson sticks, try a custom Grockit game and practice some problems for yourself!

photo credit: Mike Johnson – TheBusyBrain.com

College Admissions Strategies

jonathan peters March 10, 2010

Getting into college is a nerve-wracking task. Even after you fill out a lengthy application, have the admissions interview, and wait patiently until April, there still exists the possibility of not getting in. Not to mention that you have to continue to go to school and do well enough that college admissions officers do not raise an eyebrow when they receive your second-semester transcript. There are certain strategies, though, to make life easier for you, and hopefully to make you more likely to get into your college of choice.

Spread out the colleges. College counselors will tell you to pick three types of schools, safeties, reaches, and the in-between. Safeties are schools that are not overly competitive and that you have a good chance of getting into with your grades and standardized tests. Reaches are schools that are competitive and that may accept students with higher scores and grade points. The in-between is everything else; the schools you feel comfortable applying to, but would not be let down if you were rejected. Applying to a good mix of each is a pretty safe strategy. With the online Common Application, it is much easier to send out your documents to many schools—all through the click of a button and the payment of a small fee. Take advantage of the digital age and spread out your college choices. Trust me, you will feel much better in April.

Early decision. Schools provide this option, which allows you to apply early (October) and hear back before the end of the year. Most times, you can only apply early decision to one school, so pick wisely. You will have to commit to the school if accepted. Because schools know that you will matriculate if chosen, the acceptance rate for this round is significantly higher—sometimes by twenty percent or more. If you care about a school (perhaps your reach), this is a way to show your devotion. During the past couple of years, colleges have been scaling back on this option, so it is a good idea to get your application in before colleges entirely forego this route.

Early Action. This option is like early decision, except in most cases, the admissions result is non-binding. You can find out earlier, but you do not have to attend if chosen. Further, many schools are very clear that you can only apply early action to their school, like Yale. Other schools, such as Georgetown University, are not as strict.

Rolling Admissions. Many schools offer the option to apply as early as August or September and hear back in a few months instead of next year. This is a wise idea for those early-birds who know exactly where they want to go. If you apply early enough, your application will be that much more competitive. I applied to the University of Michigan in September of my senior year and heard back (positively) in October. I knew that if I handed in my application at the end of the cycle, I would be up against more applications and more competition, so getting the app in early was important to me.

The Waitlist. If you happen to hear back from your college of choice and they have put you on the waitlist, please do not fret. This is not the same as a rejection. You can still attend this college. Of course, the reality is they could have several hundred on the waitlist, all of whom are eventually accepted, or two and they accept one. You may never know. It is best to believe that the school is literally waiting for you. Show them you care: write a letter saying you are still interested, call the school periodically and ask about the status of your application, knit a sweater with the college’s logo on it. Basically, do anything to show that you are still interested short of disguising yourself as a student and attending classes. The admissions office is much more likely to accept someone excited about their college. Remember, the waitlist means you are accepted, the college just has to find the space. Sometimes, all it takes is waiting the school out. If an accepted student drops out in August, you still have a chance. Make sure they know that you are still interested, even after April.

Grockit Live Online Courses Are Here

brian buser March 9, 2010

We’re excited to a SAT writing lesson 1 nnounce that Grockit now offers personalized courses to students preparing for the SAT, ACT, GRE, and GMAT. Grockit courses offer a set schedule of live lessons with the best instructors in the country. Students pick the instructor and schedule that works best for them. Students and instructors are connected online and also through audio conferencing.

Each course lesson is a combination of slide based lecture and group practice questions. The curriculum adapts to each students’ strengths and weaknesses as they progress through the course material. Students also get 24/7 access to Grockit Group Study where they can work with or compete against students all over the world who preparing for the same exam. We’re so confident about the effectiveness of our courses that we’re offering a money back guarantee on score improvement.

Please select an exam to see how that Grockit courses compare to others in the industry:

SAT
ACT
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Gerunds

amanda chen March 8, 2010

A common writing mistake is thinking that gerunds are verbs. GERUNDS ARE NOT VERBS. Gerunds typically involve using a verb (in the -ing form) as a noun. Just because it looks like a verb + “ing”, does not mean it is a verb. Often times, SAT questions try to take advantage of this misconception by giving you sentences with gerunds but no verb. For example, Listening to his ipod, John skiing down the slope.

As a noun, gerunds can either be the subject or the direct object of a sentence. Let’s use the verb “to read” as an example. The gerund of “to read” is “reading” and it can be used as a subject in a sentence like
Reading Shakespeare is compulsory for every high school in America.
Or it can be used as a direct object in a sentence like
I enjoy reading Shakespeare in my English class. (Here, the verb is “enjoy” and the object is “reading”)

Now that you know how to use gerunds, you need to bear in mind three main types of mistakes involving gerunds.

Mistake 1
The first is the one mentioned in the first paragraph – the presence of a gerund but a lack of a verb. Using the previous sentence as an example, Listening to his ipod, John studying for the SAT should be Listening to his ipod, John skiis down the slope.

This mistake is a little harder to spot when there is a modifying clause that has a verb in it. For example, Sarah walking down the street, her head bent in thought. Just because there is a verb “bent” in the sentence does not make the sentence grammatical. The main clause in the sentence is “Sarah walking down the street” and this lacks a verb. The second part, “her head bent in thought” describes Sarah and is a modifier. For the sentence to be grammatical, it would have to be Sarah walked down the street, her head bent in thought.

Here are some faulty sentences. What’s wrong with them?
Many of the best kids’ movies, including both “The Labyrinth” and “The Dark Crystal,” originating in the mind of Jim Henson.
If you ignore the clause that is between the commas (“including both… Dark Crystal”), you will realize the the main sentence is “Many of the best kids’ movies originating in the mind of Jim Henson”. The problem is, this sentence needs a verb, so “originating” has to be changed to “originated”.

Shakespeare writing some of the best tragedies, such as Romeo and Juliet, which is read by just about every high school student.
As before, it is always a good habit to strip away the modifiers and identify the main part of the sentence. Romeo and Juliet is an example and “which is read by just about every high school student” is a modifier referring to Romeo and Juliet. So the main sentence here is “Shakespeare writing some of the best tragedies”. Again, this is lacking a verb so “writing” should be changed to “wrote”.

Mistake 2
The second type of mistake is a lot harder to spot unless you have a good command of English. There are some verbs that must always be followed by gerunds.

For example, admit, consider, stop, deny, appreciate are all followed by gerunds.
He admitted to having a cigarette or two during my lunch break.
Seeing as he was so lonely, I considered setting him up with a friend.
Sally stopped singing in the shower when she realized her roommates could hear her.
She denied breaking the exquisite vase on the table.
I appreciate hearing a good voice sing the national anthem.

Certain verbs are frequently used with a preposition that must also be followed by a gerund.
For example,
He complains ABOUT having to work long hours.
He insisted ON paying for the meal.
I object TO using violence to solve problems.
She looks forward TO seeing her boyfriend at Christmas.
The politician has a habit OF rewording your speeches at the last minute.

In general, it is fairly safe to assume that prepositions should be followed by gerunds.

Mistake 3
This is the “being” mistake. Being is not a verb. Try saying “I being, you being, he being, she being”. It sounds wrong doesn’t it? Being can be correctly used in the passive voice, but more often than not, being is used to create overly wordy sentences that are not the answer.

Take a look at this sentence.
Tom was punished by his teacher, not for his tardiness, but for his being extremely disrespectful towards her.

In addition to the original phrase, it is common to see a choice like “the reason being for his extreme disrespect towards her”. The presence of “being” should sound an alarm in your head immediately. Read through it quickly, and if it sounds awkward and clumsy, eliminate those choices. Go for the simplest choice such as “but for his extreme disrespect towards her”.

Check out Grockit for more SAT verbal practice with Amanda!

The SAT & the ACT: Frequently Asked Questions

vivian kerr March 1, 2010

What is the difference between the two tests?

The main difference between the two tests is the format. The SAT consists of 10 sections testing three different categories: Writing, Reading, and Math. The first section is always an essay which is part of the cumulative Writing score.

The ACT consists of 5 separate tests of varying lengths testing five different categories: English, Math, Science, Reading and Writing. The Writing section of the ACT will always be last and is optional (though highly recommended).

For both tests, each question correctly answered is worth one raw point. Unlike the SAT, there is no penalty on the ACT for marking incorrect answers on multiple-choice questions.

What do my scores mean?

On the SAT, each of the three sections has a maximum score of 800. The highest possible SAT score is a 2400. Many students aim to break the 500 mark in each section to become a competitive applicant. The most competitive scores range from 650-800.

On the ACT, you will receive a separate ACT score (from 1 to 36) for each of the five tests and a Composite ACT score, which is an average of the first four 4 tests. The composite score is also between 1 and 36. Students aim to break a score of 17 to be competitive. A score of 26 puts a student in the 90% percentile.

You will also receive subscores in English, Math, and Reading that range between 1 and 18. These scores provide you with more detail about your performance, but they are not actually used by schools. If you take the Writing Test, you will also receive a Writing Test subscore and a Combined English/Writing score.

Which test should I take?

Generall, the SAT is the preferred test for schools on the East and West coasts, while the ACT remains popular with schools in the Midwest and in the South. However, many schools accept both. To determine whether you should take the SAT, the ACT, or both, you should see what the schools to which you are applying prefer.

Many students score better on one test over the other but take both tests, determining which score to submit afterwards. The ACT Math is typically harder than the SAT Math and includes some trigonometry, while many students find the ACT English section easier than the SAT Writing questions.

You’ll want to practice both tests to get a feel for how the content is different and then choose the test that will show off your strengths. Remember also that there is no wrong answer penalty on the ACT! Even if Math and Science aren’t your strengths, you may be able to get points simply by using elimination strategies and guessing!

How do I study?

Luckily, both the ACT and the SAT test similar math, grammar, and critical reading areas, so it is entirely possible to study for both at the same time.

Working backwards from your test date, it is important to plan to study at least 30-45 minutes a day. Make flashcards for commonly tested Idioms and Vocabulary words. Memorize Math formulas that are unfamiliar to you. Keep a Strategy Sheet where you write down strategies that you run across in your reading. Practice a few different ones until you find what works best for you.

You will also want to take at least 3 full-length practice tests before the actual test. You can find these in many SAT and ACT study guides at your library or online. Most test-prep companies will schedule practice tests once a week in the final month before your test-date. Even if you are studying alone, it’s important to do this to build up your pacing ability and stamina. Don’t expect to finish your first practice test!

How do I handle stress?

The great thing about both the ACT and the SAT is that they are divided into different subject areas. Colleges will not only see your combined total score, but also how you scored in each individual subject area. For example, if you are applying as an English major and your Math scores are not as high as your Reading and Writing scores, the admissions department will certainly take that under consideration. Everyone has their strengths and weaknesses, so don’t expect yourself to score perfectly or evenly in all sections. Remember these schools want humans, not robots, and they don’t expect perfection!

To learn more about these tests or to register, check out www.act.org or www.sat.org!

ID sentence errors – adjective/adverb confusion

james kim February 26, 2010

In this segment, we will take a look at adjective/adverb confusion in the SAT writing section for the “ID sentence errors” section.

In general, an adjective is any word that is used to modify a noun or pronoun. An adverb modifies a verb, adjective or another adverb.

For example:

Adjective – The woman’s amazing voice resounded in the room.
Adverb – The woman’s voice resounded amazingly.

In the first case, the adjective “amazing” is modifying the noun “voice.” And in the second case, the adverb “amazingly” is modifying the verb “resounded.” It’s a subtle difference, but one that is tested on the SAT.

Now in everyday speech, you’re hard pressed to find someone who speaks grammatically 100% of the time. Because of this, we hear a lot of adjectives where adverbs should go. But unfortunately, the SAT doesn’t care about what happens in real life, so it’s something you’re going to have to try and spot.

Luckily, it is very simple to check. When coming across an “ID sentence errors” question, check every underlined choice to see if any of them contain an adjective. If so, then it’s matter of checking the context of the adjective.

Example: The bold spoken words could not sway the jury’s decision.

In this case, the adjective “bold” should be changed to the adverb “boldly” because it is modifying “spoken” which is an adjective. In the context of the sentence, the “words” themselves aren’t bold, but the way that they are spoken is bold.

By spotting adjectives in the ID sentence error choices, it is easy to tell if they are incorrect or not. Just remember the definitions of adjective and adverb. If the underlined adjective is modifying a noun or pronoun, then it is fine. If it is modifying an adjective, adverb or verb, then you should change it to an adverb.

Another defining factor of the adverb is that adverbs will answer the questions “how,” “when,” “where,” and “how much.” But again, make sure to keep the main tenant in mind. Adjectives – nouns/pronouns, Adverbs – verbs/adjectives/adverbs.

With these hints in mind, it’ll be easier to find the adjective/adverb questions in the ID sentence errors. Granted, not every question will deal with this subject, but if it does, it only takes a few seconds to spot whether or not an answer choice is an adjective. And, like I said before, if there is an adjective in an answer choice, then you can spot to see if it is used correctly by remembering the definitions of adjectives versus adverbs.

Play an SAT game on Grockit for more ID sentence errors practice.

Elementary Algebra Pt. 1

nicky carney February 24, 2010

Algebra is a branch of mathematics that describes equations and arithmetic “symbolically.” Whereas in normal arithmetic an expression like “2+2” has a definite answer, in algebra unidentified numbers can be symbolically represented by variables; “a,b,c,x,y, or z”. A simple addition in algebra could look like this “a+b=c”. With these concepts, we can look at an algebraic equation.

“x+6 = 16”

When we are presented with an algebraic equation like this, we can actually “solve the equation” which just means we can find the numeric value of the ‘x‘ variable in the equation. The numeric value of x that satisfies the equation, keeps the equation true, is called the solution to the equation.

So how do we start solving any equation? Well, there are a number of operations we can do on any equation and these operations are our main tools in Algebra. In Algebra, the number one concept to understand is that on equality. Any equation, or equality, can be modified by doing the same arithmetical operation to BOTH SIDES of the equation. For example, let’s take another look at the equation “x+6 = 16”. Let’s subtract a constant from both sides of the equation, in this case 6.

(x+6) – 6 = (16) – 6 On the left side of the equation, 6-6 = 0, so those two terms cancel out leaving only x. And on the right side of the equation, 16-6 = 10, so 10 remains on the right side. So after we have subtracted 6 from both sides, we are left with this equation: x = 10. Notice the form of this final equation. What is left is actually the solution to the equation! If we now know that x is equal to 6, then if we substitute 6 for x in the original equation, we’ll find that the equality is still true.

X + 6 = 16 : (10) + 6 = 16 Notice that 10 + 6 does equal 16, so we know that x = 10 is a correct solution to the equation. This was a very simple example of solving an algebraic equation. Once again, the most important concept to understand about solving algebraic equations is how to do operations on BOTH sides of the equation. The main method of solving a simple equation like this is called “isolating the variable” where on one side of the equation is only the variable term and on the other side is everything else. With this, we have the basics of elementary algebra.

Let’s try a harder problem and go through all the concepts so far:

3x + 24 = 8 + 5x : notice that there are 4 terms in the equation; 3x and 24 in the left expression, 8 and 5x in the right expression. So if we want to solve the equation, we need to isolate the variable, ‘x‘. In order to do that, we have to do arithmetical operations to each side of the equation. Since there a term with x on both sides of equation, we can do an operation to manipulate the equation so that x is only on one side of the equation. Since 3x is less than 5x, we can subtract 3x from both sides of the equation thereby leaving the equation with x on only one side of the equation.

(3x + 24) – 3x = (8 + 5x) – 3x :

3x – 3x + 24 = 8 + 5x – 3x :

24 = 8 + 2x
So now that we have only one x term on one side of the equation, we can take the next step in isolating x by subtracting 8 from both sides of the equation.

(24) – 8 = (8 + 2x) – 8 :

16 = 2x
One more step and we can completely isolate x thereby finding the solution for the equation. Let’s divide both sides of the equation by 2.

(16)/2 = (2x)/2 :

8 = x
Now we have solved the equation for x. If we go back to the original equation and substitute 8 for x, we will hopefully find that the equality still holds true.

3*(8) + 24 = 8 + 5*(8) :

24 + 24 = 8 + 40 :

48 = 48

This just about wraps up the basics of elementary algebra. In the next part I will go more in depth and develop the basic concepts discussed in this first part. For now, there are a few definitions to understand that are important for every stage of algebra from here on out.

Definitions:

term – Any number or variable, coefficient or not, in an equation or expression. Example (terms are in bold):

3x + 4 = y/2 + 10

variable – A symbol used to represent either a single number or a set of possible numbers. Example: “a,b,c,x,y,z”

expression – A set of numbers, variables, and various arithmetical operations connecting each term. Example “4x²+ 2x + 9”

equation – An equation says that two “things”, either terms or expressions, are equal to each other.

Example: “7 – c = 1 + 2c”

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