Substitution questions are some of the simplest algebra questions on the SAT. These questions provide you with an algebraic expression and give you the value of one of the variables within the equation. For example:

You might see this equation filled with variables and panic. You shouldn’t. The problem is immensely simple. Since 2y + 8x = 11, all you have to do is substitute 11 in for 2y + 8x in the expression 3(2y + 8x), and you get 3(11) = 33.

Some substitution questions are a tad more complicated. For these, you might have to do some simple math either before or after the substitution.

In this problem you have to find what 3x equals before you can substitute that value into the expression 23 – 3x. To find 3x, simply take

and add 7 to both sides, giving:

Now we can substitute that 15 into 23 – 3x:

Here we first have to solve for a by substituting 3 for b:

Once you know that a = 4, just substitute into 4a:

Occasionally the SAT will ask a word problem, and you will have to write out an expression that describes the word problem, and perhaps simplify it. For example:

To answer this question, you have to interpret the word problem. In other words, you have to figure out what is important in the word problem and how it fits into the expression you need to build. In this question, you are asked to generate an expression that describes how many cups of water there are in the bucket after Mary removes (g –3) cups. It doesn’t matter what g actually equals, because we don’t care how much water was in the bucket before Mary added g cups. The question only includes that detail to trick you. As far as we’re concerned the problem might as well have been:

To work out the equation, we take the number of cups in the bucket and subtract what was removed:

The equation to state how many cups of water are in the bucket is: total cups = f – g +3.

ETS often deliberately writes equations less clearly than it could. Instead of writing:

ETS would probably write:

The ETS writers do this simply to confuse you, which seems rather juvenile of them. But you should still be ready for it.

These types of questions usually appear in the last, difficult third of SAT math sections. If you are looking to score above a 600, you should definitely make sure that you know how to answer them.

Often the SAT will ask a question about an equation that seems impossibly complicated. In such cases, simplifying the equation can often reveal the answer more clearly or make calculating the answer a less harrowing task. There are two primary ways to simplify an equation, factoring and combining like terms.

Factoring an algebraic expression means finding factors common to all the terms in an expression and dividing them out. For example, to factor 3a + 3b, divide out the three to get 3(a + b). Factoring is merely reversing the distributive property of multiplication. Below are some examples of factoring:

Expansion involves taking a factored expression, such as 8(b + 3), and distributing one term to the other(s) by multiplying them: 8b + 24.

If an expression contains like terms you can combine those terms and simplify the equation. Like terms are identical variables that have the same exponential value.

As long as two terms have the same variable and the same exponential value, you can combine them. Note that when you combine like terms, the variable doesn’t change.

Variables that have different exponential values are not like terms and cannot be combined. Two terms that do not share a variable are also not like terms, and cannot be combined regardless of their exponential value.

A number of SAT questions will provide you with an equation such as x = yz and then ask you to show what that equation looks like in terms of y. The secret to answering such questions is a simple rule: you can perform any operation on one side of the equation as long as you perform the same operation on the other side of the equation. For the question described above, you need to isolate y from the other two variables. To do so, all you have to do is divide both sides of the equation by z.

A more difficult SAT question might ask:

To answer this question, you have to isolate x, multiply both sides of the new equation by y, and then substitute 7 for y on the right side.