Algebra | Arithmetic & algebra | Quantitative reasoning

Algebra problems generally have some equation with an unknown variable. The key ingredient to any algebra problem is to isolate the variable. Once you have the variable alone, you can determine its value. The way we isolate variables, in algebra, is to do the same operation to both sides of the equation in order to get rid of whatever is accompanying the variable. Let’s start with a simple example:

Given: $x + 5 = 10$

What is the value of $x$ ?

We can subtract 5 from both sides of the equation to cancel out the 5 on the left. By subtracting 5 from both sides, we are left with $x = 5$ .

$x + 5 x + 5 - 5 x = 10 = 10 - 5 = 5$

As long as we transform both sides of the equation by the same value, the sides will still be equal to each other.

Of course, some algebra problems are much more complicated, and it may be difficult to know what to do to isolate the variable.

To isolate an algebraic variable, perform the opposite operation on whatever numbers accompany the variable, and do the same to the other side.

For example:

Given: $3 x = 12$

What is the value of $x$ ?

The opposite of multiplication is division. So, we should divide both sides by 3.

$3 x 3 x /3 x = 12 = 12/3 = 4$

Exponents and roots might seem more complicated but they follow the same rules:

Given: $x^{2} = 100$

What is the value of $x$ ?

The opposite of squaring a number is to take the square root of the number.

$x^{2} x^{2} x = 100 = 100 = 10$

If there are many operations necessary to isolate the variable, first get rid of any values that do not have the variable attached, and save the final operation to get rid of any coefficient attached to the variable.

Given: $3 x - 3 = 27$

What is the value of $x$ ?

We’ll apply the same process in several steps:

Add 3 to both sides
Divide both sides by 3

$3 x - 3 3 x - 3 + 3 3 x 3 x /3 x = 27 = 27 + 3 = 30 = 30/3 = 10$

Sidenote

Brackets vs. parenthesis

Often questions will say something like:

$x$ is within the range [0, 10]
$y$ is within the range (0, 10)

When a range is given using square brackets, i.e. $[0, 10]$ , this means that the bounds are inclusive. The variable $x$ could have any possible value from 0 to 10, including both 0 and 10: $0 \leq x \leq 10$ .

When a range is given using parentheses, i.e. $(0, 10)$ , this means the bounds are exclusive. The variable $y$ could have any possible value between 0 to 10, but could not be 0 and could not be 10: $0 < y < 10$ .

Sometimes you’ll see them mixed:

$z$ is within the range (0, 10]

Since there is a parenthesis on the left side and a square bracket on the right, this means that $z$ could be any number greater than zero, up to and including 10: $0 < z \leq 10$ .

Understanding how to manipulate an algebra problem as we did above will be a handy skill in the GRE. Some of the questions will be much less straightforward, but you will be able to solve all of them by applying these same basic building blocks!