The SAT covers most of the same topics every single test. There are some topics that occur with greater frequency than others. With the Writing and Language Test, for example, there tends to be a higher frequency of certain types of grammar questions. If you know these simple grammar rules, you’ll get the questions correct. The same principle applies for the math and reading portions of the test.

Below is a simple toolkit for quickly increasing your overall SAT score by 100 to 200. We chose each item in the toolkit by selecting a random sample of College Board SAT tests and determining the question types that appeared most frequently. The topics that we selected, collectively, account for approximately 25% of all tests, or between 34 and 42 questions. The breakdown is as follows:

Math: 18-22 questions

Writing and Language: 10-13

Below is a list of topics in the order that they appear on this page. Scroll to the end for answers and explanations.

Mathematics

-Coordinate Geometry

-Translations

-Systems of Equations

Writing and Language

-Commas and Independent Clauses

-Comma Splices

-Verb Tense

-Most Nearly Means Questions

Mathematics

Coordinate Geometry

Slope-Intercept Form

Lines in a coordinate plane are defined by the following generic equation:

y = mx + b

· x and y define points on the line

· m is the slope

· b is the y-intercept, where the line intersects with the y axis.

As long as you understand what each part of the equation represents, then you have a powerful tool for manipulating and solving many problems that deal with lines on a coordinate plane.

Slope Formula

You can think of the slope as how steep the line is. Use the following formula to calculate the slope of a line:

m = (y2 – y1)/(x2 – x1), where (x1, y1) and (x2, y2) are points on a line.

The value of the slope itself offers very important information. Keep in mind the following scenarios:

· If the slope is positive, then the line goes on an incline.

· If the slope is negative, then the line goes on a decline.

· If the slope is positive, the greater it is, the steeper it is. The closer the slope is to 0, the more level it becomes.

· If the slope is negative, the closer it is to 0, the more level it is. The more negative it becomes, the steeper the line will be on the decline.

· Horizontal lines have a slope of 0.

· Vertical lines have an undefined slope, owing to the fact that all the x coordinates are the same, making the denominator 0—constituting an undefined fraction.

· Lines with the same slopes are parallel. E.g.: y = 5x + 2; y = 5x -17. These lines will never touch in space.

· Lines with slopes that are negative reciprocals are perpendicular. E.g. y = 2x + 3; y = -½ x – 7. These lines will intersect at exactly one point in space to form four right angles.

*Tip 1: If a line does not conform to the slope-intercept form, solve for y to get it there!

*Tip 2: Draw a coordinate plane or use your graphing calculator.

Sometimes, coordinate geometry problems can be difficult to visualize. For the trickier problems, don’t hesitate to make a quick drawing of a coordinate plane, or simply plug in all the information into your graphing calculator. Visually representing a problem is usually the first step to understanding, and ultimately, solving it.

Practice

1. A line in the xy plane passes through the point (1,3) and has a slope of -2. At what point does the line intersect with the y-axis?

A) (-5,0)

B) (5,0)

C) (0,5)

D) (-2,0)

2. Find the equation for the line who’s slope is perpendicular to y = 4x -6.

A) y = 4x

B) y = -4x + 6

C) y = -1/4x -1

D) y = x – 4

3. Two lines, a and b, intersect one another at an infinite number of points in a given coordinate plane. Which of the following statements must be true?

A) The two lines have a common slope

B) The two lines are perpendicular

C) The lines occupy three-dimensional space

D) The lines’ y-intercepts are negative reciprocals of one another.

4. A line intersects the y-axis at 4, yet never crosses the x axis. Which of the following equations will intersect this line at a 90-degree angle?

A) y = 4x – 1

B) 4x + 1 = 3x – 12

C) x = 3y + 4

D) 2x + 4 = 5y – 6

5. A line passes through the origin and has a slope of -4. Which of the following points is NOT on the line?

A) (1, -4)

B) (0, 0)

C) (13, -52)

D) (-3, -12)

Translations: Translating Words into Math

Translation problems are word problems that require you to translate words into math. In a translation problem, words signal either numbers or operations. The list below summarizes these signals.

• Is, equal to: =

• More, more than, plus, added to, sum: +

• Less, less than, difference, minus, subtract, taken from: —

• Of, times, product: X

• Quotient, divided by: ÷

• Twice: X 2

• Thrice (rare, but possible): × 3

• Greater than: >

• Greater than or equal to: ≥

• Less than: <

• Less than or equal to: ≤

• Squared; the square of a number: n2

• Cubed; cube: n3

• A number: N

For translation problems, it is usually best to translate from left to right. These problems can be quite simple, if you use this strategy. All you have to do is follow the directions and the problem will solve itself.

Example:

Three more than twice a number is eight less than the number. Find the number.

Underline all the words that can be translated, then translate from left to right:

Three more than twice a number is eight less than the number.

3 + 2 × n = n - 8

Notice that for subtraction, you have to invert the order. For the problem above, it would not make sense to subtract n from 8.

The resulting equation is: 3 + 2n = n – 8. From here all we need to do is solve for n:

3 + 2n – n = n – n – 8

3 + n = -8

3 – 3 + n = -8 – 3.

n = -11

Now, try the following translation problems on your own.

1. On weekdays, Albert spends h hours per day doing his physics homework. On each weekend day, this number is cut by two thirds. If Mileva spends one third the time as Albert studying on weekdays, but four times his amount studying on weekends, which expression indicates the number of hours that Mileva spends studying over the course of three weeks.

A) 13h

B) 5h/3 + 8h/3

C) 10h/3

D) 4h/2 + 5h/4

2. A certain shoe store has a summer sale of 30% off the original sale price, x, of each pair of shoes. Socks are 15% off. If a customer purchases exactly one pair of shoes and 2 pairs of socks (the socks are the same price), y, which expression describes the total amount of the purchase?

A) 30x + 15y

B) .7x + 1.7y

C) 2(45xy)

D) 70x + 17y

3. Susan has 10 assignments to complete before the end of the semester. Due to several absences, Sharon has x more assignments to complete than Susan. Samuel has 4 fewer assignments to complete than Sharon. If the total number of assignments that all three of them have to complete is 38, how many assignments does Samuel need to complete.

A) 17

B) 11

C) 21

D) 15

4. Union City is sponsoring a bake sale to raise money for a local charity; all proceeds will be donated to the charity. A certain vendor agrees to sell a certain number of chocolate chip cookies at 50 cents each. Another vendor agrees to sell 100 cookies more than the 1st vendor at .05 cheaper for each cookie. How many cookies do both vendors need to sell in order for them to donate a total of \$425 to the charity?

A) 150

B) 400

C) 625

D) 300

5. Ten times a number is 26 less than the number raised to the 2nd power. Find the number.

A) 36

B) -10

C) 60

D) -6

Systems of Equations

A system of equations is two or more equations linked together by two or more variables. For SAT problems, you will likely have only two equations in a system. On occasion, you may have more. As a general principle, you need just as many equations as you have variables. You have two standard methods of finding the solutions to such equations: substitutions and elimination.

Substitution involves solving for one variable, then plugging the result into the other equation in the system. When you solve the second equation, you are able to determine the value of one of the variables. You need only plug this value into either of the initial equations to determine the value of the other variable.

2x + 3y = 6; 4x + 2y = 4

I may start by solving for x in the first equation: 2x = 6 – 3y; x = (6 – 3y)/2.

I can then substitute this quotient into the second equation to solve for y:

4[(6 – 3y)/2] + 2y = 4

12 – 6y + 2y = 4; y = 2

Now you can substitute 2 in for y in either of the 2 original equations:

2x + 3(2) = 6; 2x = 0. x = 0.

The point (0,2) satisfies the system.

Elimination requires you to add or subtract the equations to eliminate one of the variables. Sometimes, in order to make this procedure work, you are required to multiply one equation through by a single factor—the factor that will achieve an equivalency between one or more terms in each equation.

E.g. 2x + 3y = 6; 4x + 2y = 4

Notice if you multiply the first equation through by a factor of 2, then you get “4x” in both equations.

2(2x + 3y = 6) --- 4x + 6y = 12.

You can now subtract the second from the first one to get rid of the 4x term: 4y = 8, y = 2. You can now plug y into either of the initial equations to find the value of x: 4x + 2(2) = 4, x = 0. The point (0,2) satisfies the system.

*For the elimination method, you may multiply each equation by a different factor in order to achieve an equivalence between one or more terms in the equations.

Now, try some problems on your own.

1. In the following system of equations, find the value of y:

2x + 4y = 6

y + x = 3

A) 3

B) 12

C) 6

D) 0

2. Line m passes through point (2,4) and has a slope of -2. At what point does line m intersect with line n, which has an equation of 3y + 6x = 12?

A) (-4, 12)

B) (2, -8)

C) The two lines never intersect

D) (6, 3)

3. If z is equivalent to the expression q + 6, and 3q is also equivalent to z – 4, find z.

A) 1

B) -5

C) 7

D) -3

4. Line g has an equation of y = 4x – 7. Line h has an equation of 2y - 8x = 16. At what point do the lines intersect.

A) Infinite number of solutions

B) No solution

C) (4, 7)

D) (7, 4)

5. Solve the following system:

3x + 6y = 12

2x + 3y = 8

A) (3, 0)

B) (0, 3)

C) (1, -3)

D) (-3, 8)

Writing & Language Test

Commas and Independent Clauses

For the SAT, there are really only three major uses of commas with which you need to concern yourselves: 1. Introductory Words and Phrases, 2. Modifiers/Descriptive Phrases, and 3. Lists and Series. You must also be familiar with a common error in comma usage: the comma splice.

1. Introductory Words and Phrases.

These features tend to demonstrate continuity between sentences. Commas almost always follow.

Example: February in New York City is often characterized by excessive cold. In addition, New York City tends to see its fair share of snow storms during this hostile month.

Notice that the phrases “in addition” expresses continuity between the ideas in the pair of sentences. Other phrases that serve this purpose are as follows:

moreover, furthermore, however, also, for example, for instance

2. Modifiers/Descriptive Phrases.

These phrases can appear at any point in the sentence. The basic rule of modifiers is that they need to be relatively close to the things that they are modifying.

A. The Beginning. If a modifying phrase occurs at the beginning of a sentence, there will be a comma immediately following it. Right after the comma should occur the subject of the sentence, or the thing that is being modified.

Example: After arriving home from school, Sue decided to walk her dog.

Notice how Sue, which immediately follows the comma, is the one who performed the action in the modifier.

B. The Middle. If a long modifying phrase occurs in the middle of a sentence, then there must be commas before and after the phrase. We call this type of modifier a parenthetical. Common usage of parentheticals for the SAT writing questions are those that disrupt the continuity between the main subject and verb.

*You can take a parenthetical out of the sentence, and that sentence will still have grammatical continuity.

Example: My brother, who is taller than I am, is not that great at basketball.

Notice that there are commas on either side of the modifying phrase. Also, notice that if you were to eliminate the modifier, the sentence would still make grammatical sense:

My brother is not that great at basketball.

*Parentheticals can also be as short as one word. Many of the same words and phrases that can serve an introductory function in a sentence may also be used as parentheticals.

Example: One can get a pretty good feel of New Yorkers if positioned in the middle of Times Square. The top of a skyscraper, however, offers a more holistic view of the City.

Notice that the word “however” can be extracted, and the sentence will still make grammatical sense.

C. The End. You can use a comma to attach a modifying phrase to a word at the end of a sentence as well.

Example: Many New Yorkers still prefer to ride the subway, despite its reputation as a colossal inconvenience.

Example 2: My sister is not nearly as talented of a painter as my brother, who happens to have been trained at one of the most acclaimed art academies in the world.

*Notice that the modifying phrase occurs immediately after the word being described. Again, placing the modifier and modified phrases close to one another is a fundamental principle of written English.

3. Lists and Series

Use commas to separate items in a list. A sequence in which each item is a complex phrase is called a series.

A. Lists. In a list of three or more items, only an “and” need separate the final two items:

Example: Common summer activities are swimming, fishing, golf and skydiving.

*As its name implies, the “Oxford comma”, a feature not uncommon in the British system, can precede the “and”.

Example: Common summer activities are swimming, fishing, golf, and skydiving.

You will never be tested on your knowledge of the Oxford comma on the SAT, however.

B. Series. When each item has a more complex structure, the sequence is known as a series. For series in which the items are really complex, especially when one or more of the items has a coordinating conjunction (e.g. and, or, but), you should use a comma in between all items:

Example: When cooking, one must be a master at multi-tasking. In the space of 10 minutes, for example, one may have to crack and beat eggs, season a piece of meat with five-to-seven herbs, and slice and boil vegetables.

Notice that the commas help make clear which activities should be grouped together. Notice also that the final comma precedes an “and”.

Comma Splices

A common error in writing is the comma splice, a variety of run-on sentence in which two independent clauses—a cluster of words that can stand alone as its own sentence—are joined only with a comma.

Example: Rain was a common occurrence on Thanksgiving, my family considered this occurrence to be ironic.

Notice that the portion of the sentence before the comma and the portion after can stand alone as their own sentences. Each part has a subject (sentence 1: “Rain”. sentence two: family.), and a verb (sentence 1: “was”. sentence 2: “considered”). To be considered complete, a sentence only needs to have these two elements; all other elements are just modifiers.

*You can correct a comma splice in two primary ways: coordinating conjunctions and punctuation.

A. Coordinating Conjunctions. Use the one that is appropriate.

For And Nor But Or Yet So

Example: Rain was a common occurrence on Thanksgiving, and my family considered this occurrence to be ironic.

B. Punctuation. Use a period, semicolon or colon to combine/separate independent clauses.

Note: For the purpose of the SAT, think of semicolons as periods. One should only use a semicolon if there is an independent clause before and after.

Example: Rain was a common occurrence on Thanksgiving; my family considered this occurrence to be ironic.

Example2: Rain was a common occurrence on Thanksgiving. My family considered this occurrence to be ironic.

Now, attempt the following problem:

1. The evolution of human beings is very much a history about the evolution of the human brain. The latest, and most complex, development is the neocortex, the part of the brain that is responsible for higher order reasoning. Many scholars even argue that the neocortex is uniquely responsible for our sense of self-awareness, species without a highly developed neocortex also lack self-consciousness.

A) NO CHANGE

B) Self-awareness—species

C) Self-awareness species

D) Self-awareness; species

2. The New York City Transit System is, arguably, one of the worst in the world. While it is notoriously quick the massive number of delays and detours can add as much as 30 minutes to one’s commute.

A) NO CHANGE

B) quick; the massive

C) quick, the massive

D) quickly massive

3. Regarding human gender, it is no longer appropriate, not to mention accurate for us to think in terms of binaries.

A) NO CHANGE

B) Accurate; for

C) Accurate, for

D) Accurate—for

Verb-Tense Errors

Be aware of any verbs that are underlined. When you see that a verb is underlined, there are only two possible errors: 1. The verb doesn’t properly agree with its subject; or 2. The verb is in the wrong tense. We will focus on the latter, as it constitutes a common error on the SAT.

To correctly answer these problems, you only need to take a look at the verbs in the sentence and the surrounding sentences to determine the time period of the occurrence in question. Next, select the answer choice that makes all the verbs consistent.

The only time that it makes sense to deviate from this rule is if there is an appropriate signal indicating events that happen in sequence.

E.g. After I finish doing my math homework, I will go to dinner with some friends.

Notice that I start with present tense and end with future tense. Here, the signal “after” allows me to combine these tenses in a way that makes sense.

Try the following question on your own:

1. Albert Einstein was not merely a scientist; he was a creative thinker. In fact, he often encouraged people to think in ways that are completely counterintuitive. In his theory of general relativity, for example, he argued that, as objects approach the speed of light, time slowed down.

A) NO CHANGE

B) time slows down

C) times have slowed down

D) time does slow down

2. tried to slow down but the ice was just too slippery. My tires did not have enough traction to prevent me from sliding into a tree. Now, my insurance company wanted to raise my rates

A) NO CHANGE

B) wants

C) has wanted

D) must want

3. Since their inception, computers have continued to speed up our lives. With the emergence of the Internet (at least for public use), we can access information much more quickly than was possible before this phenomenon’s advent. For instance, since the early 90s, the primary mode of communication between students and teachers at universities became electronic mail.

A) NO CHANGE

B) Will become

D) Has become

“Most Nearly Means”

These questions usually involve the meaning of a word or phrase in the passage. The easiest way to solve these problems is to simply substitute the answer choices into the sentence in question to see which one makes the most sense, given the context. For tougher questions, remember that the correct answer will always be directly related to the denotative meaning (dictionary definition), even as it fits the context. Consider the following schematic:

Attempt the next question on your own.

Promiscuity is considered a “dirty” word in the American lexicon. Few things are more morally objectionable than one person selfishly enjoying the carnal embrace of multiple individuals. Yet, promiscuity is not the exception, but rather the norm, if we survey the entire animal kingdom. There are many species who not only participate in such rituals casually, but there are also several who actually require this type of multiplicity as routine part of mating.

1. In line 1, the word “lexicon” most nearly means

A) mind

B) understanding

C) comprehension

D) language

Play is not a frivolous activity—at least according to many psychologists. In fact, when people play, they get necessary practice for real world activities. Children often play at the activities they will later perform as adults: baby dolls will one day become their real children and one day, they will start to pay actual mortgages on the clubhouses that they construct and manage out of Legos.

2. In context, the word “frivolous” (line 1) most nearly means

A) immoral

B) insignificant

C) sophisticated

D) serious

Clearly, he was descending—into some deep, subterranean region of his mind, through some elaborate, twisting channel that the rest of us have not the deftness, courage or aptitude to navigate. Yet, this descent, like all the others, was steady and measured, even as it was sudden, solemn and inevitable. You knew by the vacancy of his eyes, void, yet pregnant with attention.

3. In line 3, the work “pregnant” most nearly means

A) with child

B) empty

C) full

D) aware

Coordinate Geometry

1. C

2. C

3. A

4. B

5. D

Translations

1. A

2. A

3. B

4. A

5. B

6. D

Systems of Equations

1. D

2. A

3. C

4. B

5. A

Commas and Independent Clauses

1. D

2. C

3. C

Verb Tense

1. B

2. B

3. D

Most Nearly Means

1. D

2. B

3. C

Explanations

Mathematics

Coordinate Geometry

1. C

Step 1: Because this is a coordinate geometry problem, you should immediately recall the slope-intercept form: y=mx+b. Now let’s start plugging in information.

Step 2: The problem indicates that “m” or the “slope”, is -2. We can now start partially filling in values for the equation:

y=-2x + b

Step 3: We are aiming to find “b”, or the y-intercept. In order to do so, we must find the values of the other variables. We are given both “x” and “y” in the coordinate pair. Thus, we only need to fill in this last bit of information and solve for “b”.

3=-2(1) + b

3 = -2 + b

3+2=b

b = 5

The final form of our equation for line q is y = -2x + 5

Now, let’s start eliminating answer choices.

The point at which a line intersects with the y-axis would have an x coordinate of 0. Thus, we can get rid of answer choices A, B and D, as none of these meet this criteria. The only possible answer is C.

2. C.

Lines that are perpendicular have slopes that are negative reciprocals of one another. The slope of

y = 4x – 6 is 4. The negative reciprocal of 4 is -1/4. Thus, the equation that is perpendicular to the line in question must have a slope of -1/4. Answer choice C is the only one that works.

3. A.

In order for the two lines to intersect at an infinite number of points, they must overlay one another, meaning that they must have the exact same equation: they share a slope and a y-intercept. Only answer choice A fits this criteria.

4. B.

The question describes a horizontal line that crosses the y-axis at 4. The equation for this line is y = 4. For a line to cross this line at a 90-degree angle, the second line must be vertical, meaning that it will be in the form of x = a constant. Only answer choice B works. We combine like terms to get the following:

4x – 3x = -12 -1.

x = -13

5. D.

First, we must determine the equation for the line. If it passes through the origin, then the y-intercept is 0. The slope, m, is -4. Thus, the equation for the line is y = -4x. We can determine the points that lie on the line by plugging in the answer choices and seeing which ones have an equivalency on the left and right sides. All of the points work except for (-3, -12), answer choice D.

Translations

1. A

For a complex problem such as this, it is important that you not attempt to figure it out all at once. Long word problems are like hydras, multi-headed monsters. You must cut off one head at a time, so that you have a better chance at defeating it.

Let’s begin by listing what it is we know.

Let’s start with Albert’s weekdays. He studies for h hours a day.

There are 5 weekdays, so the total number of hours that he studies during the weekdays on a given week would be 5h.

Now let’s consider Albert’s weekends. We know that Albert studies 1/3 of the time that he studies on weekdays. So, for each weekend day, Albert studies 1/3 * h. There are two weekend days (Saturday and Sunday), so the expression for the amount of time that Albert studies on weekends would be 2 * 1/3 * h or 2h/3.

Now, let’s consider Mileva. She studies one-third the time that Albert spends studying on weekdays, so we need only take 5h and divide it by three. The resulting expression would be 5h/3.

On weekends, she spends four times as much time as Albert, so we have 4(2h/3) or 8h/3.

We can determine that the amount of time that Mileva spends studying for one week is all the weekdays and weekend days added together: 5h/3 + 8h/3 or 13h/3.

2. A.

To find out how much time she spends studying over the course of three weeks, we simply need to multiply the above expression by three: 3 (13h/3).

We end up with 5h + 8h which is 13h, answer choice A.

3. B.

Let’s pick this problem apart piece by piece. First, let’s start listing expressions/quantities that we know

x represents the original price of shoes

y represents the original price of socks

30% off shoes would be the original price minus 30% of the original price: x - .3x

15% off socks would be the original price minus 15% of the original price: y - .15y. We multiply this entire difference by two given that we have purchased two pairs of socks: 2(y - .15y)

The total price would be x - .3x + 2(y - .15y). When we simplify the expression we get:

.7x + 1.7y, answer choice B

4. A.

Because this is a very text-heavy problem, it’s important that you pick it apart one piece at a time. Let’s start by listing expressions for the number of assignments that each student has to complete.

Susan: 10

Sharon: x + 10

Samuel: (x + 10) – 4 or x + 6

Now, we only need to add all of these expressions together and set the sum equal to 38:

10 + x + 10 + x + 6 = 38

Now we can solve for x

2x + 16 = 38

2x = 22

X = 11

Given that we are asked about Samuel’s assignments, we need only plug in 11 into the third expression (x + 6), to get 17. Thus, the correct answer is A.

5. B

Let us begin by setting up expressions for each vendor:

First vendor: x

Second vendor: x + 100

Now let’s determine the total amount for each vendor

First vendor: .5x

Second vendor .45 (x + 100)

Given that the total proceeds must equal \$500, let’s add both expressions and set them equal to this number.

.5x + .45 (x + 100) = 425

Now let’s simplify and solve for x.

.5x + .45x + 45 = 425

.95x = 380

x = 400

6. D.

Remember to travel from left to right in the problem (except for when you are subtracting). Let’s take one term at a time.

“Ten times a number”: 10n

“Is”: =

“7 less than the number raised to the second power”: n2 – 26

The full equation is as follows: 10 = n2 – 26

Now, we only need to solve for n:

10 + 26 = n2

36 = n2

Take the square root of both sides to get

n = ±6

Because -6 is one of the options, the correct answer is D.

Systems of Equations

1. D.

When solving systems of equations, you always want to take the path of least resistance. Let’s focus on the second equation, since the variables are not attached to any coefficients. Solve for y.

x = 3 – y

Now that we have the value of x, we can plug this expression into the first equation and solve for y:

2(3 – y) + 4y = 6

Distribute the 2 and get the following equation

6 – 2y + 4y = 6

6 + 2y = 6

2y = 0

y = 0

Notice that because you are not asked for the value of x, there is no need to plug the y value in to find the value of this second variable. Don’t do any unnecessary work. You want to conserve as much time as possible. The correct answer is D.

2. A.

For this problem, we must combine our knowledge of coordinate geometry with our understanding of how to solve systems of equations.

Step 1. Find the equation for line m

We know that the line has a slope of -2, so we can start by plugging this information into the slope-intercept formula:

y = -2x + b

In order to find b, we need now only plug in the point that is given (2, 4), and solve for b.

4 = -2(2) + b

b = 8

Our equation is y = -2x + 8

Step 2: We can now set up a system of equations to determine the point at which the two lines intersect.

y + 2x = 8 (We can rearrange the terms so that x and y terms line up vertically)

3y + 6x = 12

We can divide the second equation by 3, as this number is a common factor for all terms, and get the following:

y + 3x = 4

Now let’s subtract the equations

y + 2x = 8

-(y + 3x =4)

We get -x = 4

x = -4

Plugging in -4 into the first equation (we could have chosen the second), yields:

y + 2(-4) = 4

y = 12

Thus, the point at which the two lines intersect is (-4, 12), answer choice A

3. C.

First you want to translate the wording of the problem into a system of equations:

z = q + 6

3q = z – 4

Substitution works best, given that the first equation has single-term variables. Let’s substitute the value of z in the first equation into the second equation to get:

3q = (q + 6) – 4.

Now, let’s simplify and solve for q

3q = q + 2

2q = 2

q = 1

Now that we have the value of q, we can plug it into either equation to find the value of z. Let’s plug into the first equation, as it is the simplest:

z = 1 + 6

z = 7, answer choice C

4. B.

Remember to always put lines in standard slope-intercept form, y = mx + b. Doing so will give you fundamental information about the behavior of lines. The first lines is already in slope-intercept form. We only need to transform the second line:

Divide each term by 2, which will yield y – 4x = 8. Adding 4x to both sides yields the following:

y = 4x + 13.

The two lines have the same slope, meaning they run parallel. Lines that run parallel will never intersect, so the system has no solution, answer choice B.

5. A.

We want to manipulate one of the equations so that we can subtract and eliminate one of the terms. The second equation is more amenable to manipulation:

3x + 6y = 12

-2(2x + 3y = 8)

3x + 6y = 12

-4x + -6y = -16

We can now add the two equations together to eliminate the y term.

-x = -4

x = 4

Now that we have found the value of x, we can plug it into one of the equations to find the value of y. Let’s plug into the second equation:

2(4) + 3y = 8.

8 + 3y = 8

3y = 0

y = 0

The point (3, 0) satisfies the system—answer choice A.

Writing and Language Test

Commas and Independent Clauses

1. D.

Before the comma, we can see that we have a full independent clause. There is a subject (“scholars”) and a verb (“argue”). There is even a dependent clause that extends from “that” all the way to “awareness”. Looking beyond the comma, we see that there is another independent clause, with a subject (“species”) and a verb (“lack”). Thus, we have two separate sentences. Because these sentences are joined only with a comma, we have a classic comma splice. We can resolve this situation two ways

-By adding a coordinating conjunction. However, there are no answer choices that provide this resolution.

-By separating the sentences with either a period or semi-colon.

Answer choice D is the only one that works.

2. C.

The beginning of the sentence, “While it is notoriously quick and massive”, qualifies as an introductory phrase. Thus, a comma should follow the word “quick”. The correct answer is C.

3. C.

The phrase, “not to mention accurate”, breaks up the continuity of the sentence. If you were to remove this part of the sentence, you would still have a complete, grammatical thought. Thus, we need a comma after accurate, to make the phrase an appropriate parenthetical. Given that we need punctuation, A will not work. B. will not work because “for” to “binaries” is not a complete sentence. D. could work if there were a dash after “appropriate”, thereby keeping the punctuation consistent. The correct answer is C.

Verb Tense

1. B.

Although we are discussing a historical figure from the past, in this particular sentence we are articulating a general rule of relativity theory. Notice that the first verb used to describe the rule, “approach”, is in present-tense. Thus, we need a present-tense verb to match. B is the only answer choice that works.

2. B.

The key signal in the sentence in question is the word “Now”. This word indicates that we are referencing the present moment, a continuing condition. Thus, we need an answer choice that places us in the present tense. B is the only answer that works.

3. D.

The word “since” is very important to verb-tense questions. “Since” indicates that an action began some time in the past and continues to the present. The tense that is most appropriate here is the present-perfect, which uses “has” or “have” (depending on whether or not the accompanying noun is singular or plural) for its conjugation. Thus, D is the correct answer.

Most Nearly Means Questions

1. D.

If we plug in all the choices, we find that all of them make sense. However, there are clues that would allow us to eliminate some definitively. Take B and C, for example. They have nearly the same meaning. Because there cannot be two correct answers, we can eliminate both of them. Now, let’s examine A and D. In the first sentence, there is an important clue, the term “word”. Thus, the sentence is likely pointing to concrete features of the English language, rather than something as abstract as the “mind”. We can, therefore, eliminate A. The correct answer is D.

2. B.

The paragraph discusses the importance of play for children to learn how to be adults. Thus, the word that works best is a word that means the opposite. A “frivolous” activity would be one that is not important. A, C nor D means “unimportant”. The word we need is “insignificant”, or answer choice B.

3. C

The key word in the sentence is “yet”. The sentence tells us that the protagonist’s eyes were “vacant, yet pregnant”. The word “pregnant” has to mean the opposite of “vacant”. We know that the passage does not discuss the biological condition of pregnancy, so we can eliminate A. B is a synonym for “void”, so we can also eliminate this answer choice. D would be awkwardly redundant (“aware with attention”). Thus, the most befitting answer choice is C, “full”, which is the opposite of “vacant”.