A linear equation is any equation where the highest power of the unknown, which I shall call x, is 1.  To illustrate more clearly with a few examples:

x+1 = 4; 10x = 3; x = 18 – 4x are three examples of linear equations

x² + 2 = 2x and x³ = 8 are not linear equations because there are x’s that are raised to a higher power than 1.

A linear equation with 1 variable is the simplest type to solve.  There is 1 equation and 1 unknown, which means that the unknown can always be determined.  To solve such an equation, you need to rearrange the equation to have like terms on either side of the equal sign.  Put another way, you are trying to isolate x (or whatever the variable is called) on one side of the equation.

For example, if 2x = 234, to isolate x, we have to divide the entire equation by 2.  Doing this, we get x = 117.

If there are x’s and numbers on either side of the equal sign, we add and subtract values to isolate x on one side.  Suppose 2x – 17 = 18 – 3x

The first thing we could do is to add 17 to both sides to get: 2x – 17 + 17 = 18 – 3x + 17

This reduces to 2x = 35 – 3x

Now, we need to have all the x’s on one side so we add 3x to both sides to get: 2x + 3x = 35 – 3x + 3x

This reduces to 5x = 35

Dividing by 5 on both sides, we get x = 7

What I just went through was a fairly simple algebraic equation.  The questions on the GMAT will look more complicated but you are essentially doing the same thing: manipulating both sides of the equation in the same way to isolate x.  Let’s try a practice problem from Grockit.

To tackle this question, we multiply both sides by 2+3/x to get 3 = 2+3/x. .  (This is also known as cross multiplying where in general if a/b-c/d,

then ad = bc

To simplify 3 = 2+3/x,

we multiply the entire equation by x to get 3x = 2x + 3.  This leaves you with a much simpler equation that you already know how to solve.

What’s a little trickier than manipulating algebraic equations is translating a word problem into an algebraic equation.  Here’s another practice problem:

Jack and his brother are sharing a monster piece of licorice that is 28 inches long. Since Jack is older, his share is 8 inches longer than his brother’s. How long, in inches, is Jack’s brother’s piece?

The way to solve this problem is to let something be x.  Here’s what happens if we let Jack’s piece be x inches.

Jack’s piece = x inches

Jack’s brother’s piece = x – 8 inches

Total length of licorice = Jack’s piece + Jack’s brother’s piece = 28 = x + (x-8)

This means that x = 18 inches.  But remember that the question wants the length of Jack’s brother’s piece, which we have defined as x – 8.  So the correct answer is 10 inches.

Here’s what happens if we let Jack’s brother’s piece be x inches.

Jack’s brother’s piece = x inches

Jack’s piece = x + 8 inches

Total length of licorice = 28 = x + (x+8) and we determine that x = 10.  In this case, since we have already defined Jack’s brother’s piece to be x, there is no further step we need to take.

In general, here are a few things to keep in mind.

• if there is only one unknown, you only need one equation to determine the value of the unknown
• in dealing with algebraic equations, remember that anything you do to one side (be it adding, subtracting, multiplying or dividing) you need to do to this other side too.
• in dealing with word problems, define something to be x and see if you can define other things in terms of x only.  (For example, in the question about licorice, you would not want to let Jack’s piece be x inches and his brother’s be y inches)  Don’t introduce unnecessary variables if it can be expressed in terms of an existing variable.

A linear equation is any equation where the highest power of the unknown, which I shall call x, is 1.  To illustrate more clearly with a few examples: x+1 = 4; 10x = 3; x = 18 – 4x are three examples of linear equations x² + 2 = 2x and x³ = 8 are not linear equations because there are x’s that are raised to a higher power than 1. A linear equation with 1 variable is the simplest type to solve.  There is 1 equation and 1 unknown, which means that the unknown can always be determined.  To solve such an equation, you need to rearrange the equation to have like terms on either side of the equal sign.  Put another way, you are trying to isolate x (or whatever the variable is called) on one side of the equation. For example, if 2x = 234, to isolate x, we have to divide the entire equation by 2.  Doing this, we get x = 117. If there are x’s and numbers on either side of the equal sign, we add and subtract values to isolate x on one side.  Suppose 2x – 17 = 18 – 3x The first thing we could do is to add 17 to both sides to get: 2x – 17 + 17 = 18 – 3x + 17 This reduces to 2x = 35 – 3x Now, we need to have all the x’s on one side so we add 3x to both sides to get: 2x + 3x = 35 – 3x + 3x This reduces to 5x = 35 Dividing by 5 on both sides, we get x = 7 What I just went through was a fairly simple algebraic equation.  The questions on the GMAT will look more complicated but you are essentially doing the same thing: manipulating both sides of the equation in the same way to isolate x.  Let’s try a practice problem from Grockit. To tackle this question, we multiply both sides by 2+3/x to get 3 = 2+3/x. .  (This is also known as cross multiplying where in general if a/b-c/d, then ad = bc To simplify 3 = 2+3/x, we multiply the entire equation by x to get 3x = 2x + 3.  This leaves you with a much simpler equation that you already know how to solve. What’s a little trickier than manipulating algebraic equations is translating a word problem into an algebraic equation.  Here’s another practice problem: Jack and his brother are sharing a monster piece of licorice that is 28 inches long. Since Jack is older, his share is 8 inches longer than his brother’s. How long, in inches, is Jack’s brother’s piece? The way to solve this problem is to let something be x.  Here’s what happens if we let Jack’s piece be x inches. Jack’s piece = x inches Jack’s brother’s piece = x – 8 inches Total length of licorice = Jack’s piece + Jack’s brother’s piece = 28 = x + (x-8) This means that x = 18 inches.  But remember that the question wants the length of Jack’s brother’s piece, which we have defined as x – 8.  So the correct answer is 10 inches. Here’s what happens if we let Jack’s brother’s piece be x inches. Jack’s brother’s piece = x inches Jack’s piece = x + 8 inches Total length of licorice = 28 = x + (x+8) and we determine that x = 10.  In this case, since we have already defined Jack’s brother’s piece to be x, there is no further step we need to take. In general, here are a few things to keep in mind.

• if there is only one unknown, you only need one equation to determine the value of the unknown
• in dealing with algebraic equations, remember that anything you do to one side (be it adding, subtracting, multiplying or dividing) you need to do to this other side too.
• in dealing with word problems, define something to be x and see if you can define other things in terms of x only.  (For example, in the question about licorice, you would not want to let Jack’s piece be x inches and his brother’s be y inches)  Don’t introduce unnecessary variables if it can be expressed in terms of an existing variable.

One of the hallmark points of confusion on the GMAT is the dreaded Yes/No Data Sufficiency question.  In a Value question, such as “What is the value of x?” the question of sufficiency is a familiar one: if you can solve for x, you have sufficiency.  But in a Yes/No question, especially when variables are involved, finding a solid answer can be a much cloudier process.

The best way to clear this fog is with a concrete example.  Let’s look at this Data Sufficiency question, along with its first statement:

Is x positive?

(1) x^2 > 1.

Is Statement (1) sufficient to answer the question?  Unless you have a comprehensive understanding of the underlying Number Properties at work here, your first reaction to this statement is likely to try out different numerical values for x, because working with real numbers instead of variables will be a much more comfortable place for most of us.  We are free to try out any value for x, but our first consideration in checking this statement should be that the number we pick is permissible, according to the statement.  If it is not, we can’t even consider the number as an example.

Is zero a permissible number to use here?  Well, if x = 0, then x^2 is also 0, and this statement tells us that x^2 has to be greater than 1.  You have to take the statements as true, so zero is NOT a number we can use here (not permissible).

How about x = 2?  That puts us into permissible territory, because 2^2 = 4, and 4 > 1.  But even that is only half the battle.  Now that we know x = 2 is a permissible example, we have to see what answer it yields to our original question, “Is x positive?”  Since 2 is a positive number, the answer here is “Yes.”

Now we have one example in the bank, and we know that, given the information in this statement, the answer to the question can be “Yes.”  But is that enough to declare sufficiency?  Unfortunately, it is not.  If this statement is sufficient to answer the question, it will give us an unequivocal Yes or No answer; we know now that the answer could be Yes, but could it also be No?

Well, if the answer could be No, then that would mean x could be negative or zero.  We’ve already seen that x can’t be zero (because it’s not permissible, remember?), so what about x being negative?  Let’s take the flip side of our other example and try x = -2.  It would certainly answer the question stem with a No, but is it permissible?

Remember, the statement mandates that x > 1.  Working a little calculation, (-2)^2 equals (-2)(-2), and since the product of two negative numbers is a positive number, x^2 = 4 when x = -2.  So our second example is permissible after all, and it answers the question “Is x positive?” with a resounding “No.” Since we have answered the question with a potential “Yes” (when x = 2) and a potential “No” (when x = -2), this statement is actually insufficient in the end; we require further information to determine whether or not x is positive.

As we see, it is absolutely necessary to remember what must be assumed as true (the statements) and what may or may not be true (the question stem) when Picking Numbers in these types of problems.  While this specific example is not as challenging as some, and you may have logically thought through it with number properties rules from the outset, this thought process is vital to learn for these types of questions, and will be most helpful with the most challenging questions, where you cannot gather the potential scenarios quickly at a glance without doing some scratch work.  Questions like this are exactly why we’ve kept such a close watch on the methods used in Data Sufficiency questions throughout the new GMAT revision.  When the Yes/No monster rears its ugly, convoluted head, never forget when picking numbers: First permissible, then Sufficient!

Questions involving rectangular solids, particularly data sufficiency questions, test whether you understand the concept of volume and surface area.   You essentially need to remember that you need three different values to find volume and surface area (the length, the width and the height).  If the prompt and statements 1 and 2 are lacking some these values or some way to find them, neither of the statements will be sufficient. A rectangular solid is formed by 3 pairs of similar rectangular faces.  In other words, 6 rectangular faces in total.

The formulas you need to remember for a rectangular solid are

Volume = Length (l) x Width (w) x Height (h)

Surface Area = (2 x Length x Width) + (2 x Length x Height) + (2 x Width x Height)

If length = width = height, that means that the rectangular solid is, in fact, a cube. Other vocabulary that might be important is the terms vertex and edge. A vertex is a mathematical way of referring to the corner of any figure.  The rectangular solid above has 8 vertices (plural of vertex), can you identify them?  The edge is simply the lines you see in the diagram above: the line where two surfaces meet. Questions involving cylinders are similar and perhaps easier because there are only two values you need to know to solve cylinder-problems – the radius (r) and the height (h). If you don’t know the radius, anything that enables you to determine the radius, such as the diameter (radius = diameter / 2) or the circumference (radius = circumference / 2pi) will suffice. Regarding cylinders, the formulas you need to know are

Volume = area of the base circle x height = pi x (radius)² x height

Surface Area = (2 x pi x (radius)² )+ (pi x (diameter) x height)

Let’s try a problem: A cylindrical water tank has a stripe painted around its circumference, as shown in the figure provided. What is the surface area of this stripe? (1) y = 0.7 (2) The height of the tank is 2 meters. To find the surface area of the stripe, you need to know the circumference of the cylinder, but there is not data in the question that gives you the radius or diameter to let you find the circumference.  Hence the answer should be that neither statement together is sufficient.

GMAT data sufficiency questions test your ability to analyze a quantitative problem and recognize which information is necessary to figure out the solution.  What a data sufficiency question does NOT test you on is your ability to calculate and number-crunch.  A simpler way of addressing this might be to ask yourself a question as you work through a data sufficiency problem: “Is this enough?”  Keep this in mind as you evaluate (and on test day, avoid) two specific common errors that test-takers make while taking the GMAT:

Mistake #1: Combining statements when unnecessary

This is done when a test-taker looks at both statements and says “Yes, if I have both pieces of information, then I can figure out the answer, so together the statements are sufficient.”  However, you must remember that you’re also asked if either statement ALONE is enough to answer the question.  Understanding the differences among all five answer choices in itself can be a boost to your quantitative score.  As you look at each individual statement, ask yourself, “is this enough?”  Once you can definitively answer yes or no, you are then closer to an answer to the data sufficiency problem.

Mistake #2: Over-calculating

Since you may not need to calculate an actual value for a data sufficiency question, you should avoid going into the calculation step unless absolutely necessary.  For example, if dealing with a statement like:

2x + 15 – 7x + 3² = -1

Instead of trying to plow through with the calculations as you might have to do in a problem-solving question, recognize that you have one variable in this equation (x), and that this is solvable.  So if this shows up in a data sufficiency question, the answer to the question “is this enough?” is yes, and again you are closer to solving your data sufficiency question.

Though data sufficiency questions look very abstract, there’s a hidden beauty involved in solving them.  Practice these while taking on the mindset of “is this enough?” to maximize your time-management ability for the GMAT.