Algebra | Arithmetic & algebra | Quantitative reasoning

Algebra problems usually give you an equation with an unknown variable. Your main job is to isolate the variable - that is, get the variable by itself on one side of the equation. Once the variable is alone, you can read off its value.

To isolate a variable, you apply the same operation to both sides of the equation. This keeps the two sides equal while you remove whatever is attached to the variable.

Let’s start with a simple example:

Given: $x + 5 = 10$

What is the value of $x$ ?

The variable $x$ has $+ 5$ attached to it. To undo “add 5,” you subtract 5. Subtract 5 from both sides to cancel the 5 on the left.

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

As long as you transform both sides of the equation in the same way, the two sides will remain equal.

Of course, some algebra problems are more complicated, and it may not be immediately obvious what to do first.

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 $x$ ?

Here, $x$ is multiplied by 3. The opposite of multiplication is division, so divide both sides by 3.

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

Exponents and roots follow the same idea:

Given: $x^{2} = 100$

What is the value of $x$ ?

Here, $x$ is squared. The opposite of squaring is taking the square root, so take the square root of both sides.

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

If there are several operations, it helps to work in a sensible order: first remove any terms that don’t include the variable, and save the final step for removing 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 useful on the GRE. Even when a question is less straightforward, you can still solve it by applying these same basic building blocks.

Algebra