Linear Algebra - SAT Math
Card 0 of 20
The expression 
is equivalent to which of the following?
The expression 
If you found yourself staring at the initial fraction with no idea how to get started on the algebra, you’re not alone. The first big lesson here is that you should always take a look at the answer choices before you get started. The SAT involves a lot of “algebraic equivalency” – problems that provide you with an algebraic expression and ask you which answer choice is equivalent to it – and as you can see in this case, the answers aren’t necessarily any simpler or cleaner than the original. So an important concept when you’re translating algebra is to see which options they give you for the translation. That way you have a goal in sight and aren’t just casually performing algebra steps in the hopes of arriving at an answer choice.
Then keep in mind: with algebraic equivalency, that equivalency has to hold for all values of the variable. They’re not asking you “what is y?” but rather “which algebraic expression equals this one?” So algebraic equivalency problems – those with variables in the answer choices – are fantastic opportunities to just pick numbers and see which answer choice holds true.
For example, here if you decided to try 
, then the initial equation would be 
. Now your job would be to plug in 
to the other answer choices to see if you get a match at 
.
For 
, clearly you won't get a fraction by plugging in 
so that is incorrect. For 
you should also see quickly that the answer is not 
. That leaves the two similar-looking fractions, 
and 
. If you plug in 
to 
you'll get 
. Since 
, this choice works out to exactly 
, proving that you have the right answer.
If you found yourself staring at the initial fraction with no idea how to get started on the algebra, you’re not alone. The first big lesson here is that you should always take a look at the answer choices before you get started. The SAT involves a lot of “algebraic equivalency” – problems that provide you with an algebraic expression and ask you which answer choice is equivalent to it – and as you can see in this case, the answers aren’t necessarily any simpler or cleaner than the original. So an important concept when you’re translating algebra is to see which options they give you for the translation. That way you have a goal in sight and aren’t just casually performing algebra steps in the hopes of arriving at an answer choice.
Then keep in mind: with algebraic equivalency, that equivalency has to hold for all values of the variable. They’re not asking you “what is y?” but rather “which algebraic expression equals this one?” So algebraic equivalency problems – those with variables in the answer choices – are fantastic opportunities to just pick numbers and see which answer choice holds true.
For example, here if you decided to try 
For 
Compare your answer with the correct one above
If 
, which of the following is equivalent to 
?
If 
When the SAT asks "equivalent expression" questions like you see here, it is almost always faster and easier to pick numbers to test the answer choices against the original; the algebra can be time-consuming and a bit abstract, but since two equivalent expressions will produce the same number in both forms, you can get away with testing numbers.
When you do pick numbers, it's best to pick numbers that are easy to calculate. Here you might pick 
so that you can easily set some of the 
denominators in the answer choices equal to 1, making for quicker arithmetic. If you do that, you'll find that the original expression works out to:
Now your job is to test the answer choices using 
to see which answer choice produces the value 
. And in doing this work, you should find that only 
gives you that 2 you're looking for. When you plug in 
this expression becomes:
which works out to 
.
When the SAT asks "equivalent expression" questions like you see here, it is almost always faster and easier to pick numbers to test the answer choices against the original; the algebra can be time-consuming and a bit abstract, but since two equivalent expressions will produce the same number in both forms, you can get away with testing numbers.
When you do pick numbers, it's best to pick numbers that are easy to calculate. Here you might pick 
Now your job is to test the answer choices using 
Compare your answer with the correct one above
Which of the following expressions is equivalent to 
?
Which of the following expressions is equivalent to 
When the SAT asks you "equivalent expression" questions, it is often much easier to plug in numbers than it is to try to recreate the abstract algebra. And this strategy works because if two expressions are truly equivalent, then when you plug in numbers for variables you'll get the same answer.
The technique here is to pick an easy-to-calculate number to plug in for the variable, and then to get a numerical value for the given expression. Then you can plug in the same number as the variable for each answers, and see which choice(s) match the output value.
Here you might pick 
, making for an easy number to calculate with. That makes the value of the expression 
When the SAT asks you "equivalent expression" questions, it is often much easier to plug in numbers than it is to try to recreate the abstract algebra. And this strategy works because if two expressions are truly equivalent, then when you plug in numbers for variables you'll get the same answer.
The technique here is to pick an easy-to-calculate number to plug in for the variable, and then to get a numerical value for the given expression. Then you can plug in the same number as the variable for each answers, and see which choice(s) match the output value.
Here you might pick 
Compare your answer with the correct one above
If 
and 
. then the expression above is equivalent to which of the following?
If 
Whenever you encounter a multi-denominator expression, simplify that expression by multiplying the top and bottom by the least common multiple of the different denominators. Here the least common multiple of 
and 
is simply 
. With that multiplication you see that you are left with:
Next you should factor the 
in the numerators so that you can leverage the fact that 
. With this factoring and then substituting 
, you see that:
Whenever you encounter a multi-denominator expression, simplify that expression by multiplying the top and bottom by the least common multiple of the different denominators. Here the least common multiple of 
Next you should factor the 
Compare your answer with the correct one above
Which of the following expressions is equivalent to 
?
Which of the following expressions is equivalent to 
When you approach algebra that features multiple denominators - as you see in this problem, you begin with four "levels" of fraction - a strong algebraic first step is to "multiply by one." This means that you create a fraction with the same numerator and denominator, and use it to cancel all the smaller denominators and greatly reduce the number of fraction "levels" you're working with.
Here, for example, note that the inner fractions include denominators of 
If you create a fraction of 
to multiply by, that's the same thing as multiplying by 1 (and therefore keeping the value the same), and it will cancel several of the denominators that are making the given fraction complicated:
Now, while this does not match an answer choice yet, it's vastly streamlined compared to the original. And it also lends itself to factoring. The numerator is a classic Difference of Squares setup: 
. And the denominator has a common 
in each term that can be factored. So you can make your fraction look like:
Note that the 
terms will cancel, leaving you with 
, the correct answer.
When you approach algebra that features multiple denominators - as you see in this problem, you begin with four "levels" of fraction - a strong algebraic first step is to "multiply by one." This means that you create a fraction with the same numerator and denominator, and use it to cancel all the smaller denominators and greatly reduce the number of fraction "levels" you're working with.
Here, for example, note that the inner fractions include denominators of 
Now, while this does not match an answer choice yet, it's vastly streamlined compared to the original. And it also lends itself to factoring. The numerator is a classic Difference of Squares setup: 
Note that the 
Compare your answer with the correct one above
Given that 
, the expression 
is equivalent to which of the following?
Given that 
When the SAT asks you to find an equivalent expression, it is often fastest to pick numbers. To do so, choose a number for each variable in the given expression, making sure to choose easy numbers to work with for the situation. Here you know that 
cannot be 1 or -1, so you might pick a small number like 
. Once you've identified your numbers for the given expression, plug those in and get a target value. Here that's:
Now you can plug 
into all the answer choices, looking for a match; after all, if the expressions truly are equivalent, then they will produce the same value given the same input for 
. Note that you can stop doing any calculation as soon as you realize you won't get your target value. For example, with 
you know you won't end up with an improper fraction when multiplying 
by 
, so you don't actually have to perform the math if you know it won't match.
In doing so here, you'll find that the only match is 
, as 
.
When the SAT asks you to find an equivalent expression, it is often fastest to pick numbers. To do so, choose a number for each variable in the given expression, making sure to choose easy numbers to work with for the situation. Here you know that 
Now you can plug 
In doing so here, you'll find that the only match is 
Compare your answer with the correct one above
The expression 
is equivalent to which of the following?
The expression 
Often when you're dealing with equivalent expression questions on the SAT, the fastest way to an answer is to pick numbers. That means choosing a number for each variable in the given expression, plugging that number in, and establishing a target value. Then you can plug in that same number into each answer choice and see which choice(s) match the original. If the expressions are equivalent, the target value will match.
Here you might try 
, which would make the target value calculation look like:
Which simplifies quickly to 
If you then plug 
into the answer choices, you'll see that only 
returns the target value of 
:
Often when you're dealing with equivalent expression questions on the SAT, the fastest way to an answer is to pick numbers. That means choosing a number for each variable in the given expression, plugging that number in, and establishing a target value. Then you can plug in that same number into each answer choice and see which choice(s) match the original. If the expressions are equivalent, the target value will match.
Here you might try 
Which simplifies quickly to
If you then plug 
Compare your answer with the correct one above
The expression 
is equivalent to which of the following?
The expression 
When you face "equivalent expression" questions on the SAT, often the quickest way to get an answer is by picking numbers. That means choosing a number for each variable in the given expression, then plugging those numbers in to elicit a target value. When you then plug your numbers in for the variables in the answer choices, the correct answer will produce the same target value. This is because if the expressions really are equivalent, they'll produce the same outcome for the same input.
Here you might choose to use 
, as your number-picking goal should always be to choose a number that's easy to work with. If you do so, you would get an initial expression of:
When you then apply 
to the answer choices, you'll find that only one choice, 
, returns the same target value of 
:
Therefore 
is correct.
When you face "equivalent expression" questions on the SAT, often the quickest way to get an answer is by picking numbers. That means choosing a number for each variable in the given expression, then plugging those numbers in to elicit a target value. When you then plug your numbers in for the variables in the answer choices, the correct answer will produce the same target value. This is because if the expressions really are equivalent, they'll produce the same outcome for the same input.
Here you might choose to use 
When you then apply 
Therefore 
Compare your answer with the correct one above
The expression 
is equivalent to which of the following?
The expression 
When you face "equivalent expression" questions on the SAT, often the quickest way to get an answer is by picking numbers. That means choosing a number for each variable in the given expression, then plugging those numbers in to elicit a target value. When you then plug your numbers in for the variables in the answer choices, the correct answer will produce the same target value. This is because if the expressions really are equivalent, they'll produce the same outcome for the same input.
Here you might choose to use 
as your number-picking goal should always be to choose a number that's easy to work with. If you do so, you would get an initial expression of:
Now your job is to plug in that same 
into each answer choice to see which one produces the target value of 
. In doing so, you'll find that the only one that does is 
, as that produces 
.
When you face "equivalent expression" questions on the SAT, often the quickest way to get an answer is by picking numbers. That means choosing a number for each variable in the given expression, then plugging those numbers in to elicit a target value. When you then plug your numbers in for the variables in the answer choices, the correct answer will produce the same target value. This is because if the expressions really are equivalent, they'll produce the same outcome for the same input.
Here you might choose to use 
Now your job is to plug in that same 
Compare your answer with the correct one above
Which of the following is equivalent to 
?
Which of the following is equivalent to 
This problem offers an incredible shortcut for those who check the answer choices before performing algebra. The given expression is a proper fraction, in which the numerator (
is between 4 and 5) is smaller than than the denominator (which is 6). Of the answer choices, ONLY ONE follows that same format, 
. The others all have a numerator which is greater than the denominator, so none can be correct.
To work on these choices algebraically, a good strategy is to try to make each answer choice look like the given expression. You can see this with the correct answer. Given 
and 
, how would you make one look like the other? Multiplying both numerator and denominator by 
would mean that you take the answer choice numerator, 
and convert it to 
. And since if you multiply by the same numerator as denominator you're multiplying by one, you're allowed to do that. So you could do:
And since 
, you can simplify the denominator and see that you've arrived at the given expression:
This problem offers an incredible shortcut for those who check the answer choices before performing algebra. The given expression is a proper fraction, in which the numerator (
To work on these choices algebraically, a good strategy is to try to make each answer choice look like the given expression. You can see this with the correct answer. Given 
And since 
Compare your answer with the correct one above
If 
, what is the value of 
?
If 
The first step on this problem is to take the equation you're given, 
, and solve for 
. To do that, divide both sides by 
to arrive at 
. Importantly, recognize that the question is not asking for the value of 
, but rather for the value of the expression 
. To find that answer, plug in 
and the expression will look like 
. This then equals 
, making the correct answer 
.
The first step on this problem is to take the equation you're given, 
Compare your answer with the correct one above
If 
, then what is the value of 
?
If 
Whenever you're solving for a variable in an equation that has fractions, it's a good first step to multiply both sides by the denominator to get all the variables and values outside of the fractions. Here that means. multiplying both sides by 
to get 
.
Next, you have a linear equation in which the variable has a coefficient, so divide by the coefficient to isolate the variable. If you divide both sides by 
, you then have: 
. You can then reduce the fraction 
to 
, and since the answer choices only use integers and decimals, you can convert the fraction to the proper decimal, 
Whenever you're solving for a variable in an equation that has fractions, it's a good first step to multiply both sides by the denominator to get all the variables and values outside of the fractions. Here that means. multiplying both sides by 
Next, you have a linear equation in which the variable has a coefficient, so divide by the coefficient to isolate the variable. If you divide both sides by 
Compare your answer with the correct one above
If 
, what is the value of 
?
If 
The first step on this problem is to use the equation that you're provided so that you can solve for 
. To do so, take 
and divide both sides by 
. That leaves you with 
. Of course, the question doesn't ask you for the value of 
, but rather the value of 
. So plug in 
and the expression looks like: 
which is 
. That simplifies to 
, giving you your final answer.
The first step on this problem is to use the equation that you're provided so that you can solve for 
Compare your answer with the correct one above
If 
, what is the value of 
?
If 
Note that this problem does not ask you to solve for a particular variable, but instead a particular fraction or ratio, 
. For that reason, your goal as you perform algebra should be to isolate the ratio of 
and 
. You can do that by multiplying both sides by 
. That gives you:
, which simplifies to 
.
You can then just flip each fraction (in doing so, you're doing the same thing to both sides, namely taking the reciprocal), to get:
, which simplifies to 
.
Note that this problem does not ask you to solve for a particular variable, but instead a particular fraction or ratio, 
You can then just flip each fraction (in doing so, you're doing the same thing to both sides, namely taking the reciprocal), to get:
Compare your answer with the correct one above
If 
and 
, what is the value of 
?
If 
Whenever the SAT asks you to solve for the value of a variable, you have two options: you can either use your answers as assets and backsolve, or you can solve for the variable directly. In this case, both are equally good options, and you should use whichever method is most comfortable. Generally, however, solving directly is usually the best option on the SAT non-calculator section.
If you substitute the value 
into the equation given, it becomes:
Then you can decide whether to solve directly or plug in the answer choices. If you solve directly, then multiply both sides by 
to eliminate the denominator, yielding:
Your final step is then to divide both sides by 
and you have 
.
If you were to plug in the answer choices, you would find that only 
would work, as 
.
Whenever the SAT asks you to solve for the value of a variable, you have two options: you can either use your answers as assets and backsolve, or you can solve for the variable directly. In this case, both are equally good options, and you should use whichever method is most comfortable. Generally, however, solving directly is usually the best option on the SAT non-calculator section.
If you substitute the value 
Then you can decide whether to solve directly or plug in the answer choices. If you solve directly, then multiply both sides by 
Your final step is then to divide both sides by 
If you were to plug in the answer choices, you would find that only 
Compare your answer with the correct one above
If 
, what is the value of 
?
If 
The first key to this problem is to combine like terms; each side of the equation has multiple 
terms for you to sum to simplify the equation. Your streamlined equation should look like:
Now you can get all the 
terms on one side and all the numeric terms on the other so that you're ready to solve. That means subtracting 
and 
from each side, yielding:
Then you can divide both sides by 
to arrive at your final answer:
The first key to this problem is to combine like terms; each side of the equation has multiple 
Now you can get all the 
Then you can divide both sides by 
Compare your answer with the correct one above
If 
, then what is the value of 
?
If 
To solve this problem algebraically, take the given equation and subtract 
from both sides to isolate the 
term. That gives you:
Then you can divide both sides by 
to yield:
When you plug in 
to 
, that gives you 
, which is 
.
To solve this problem algebraically, take the given equation and subtract 
Then you can divide both sides by 
When you plug in 
Compare your answer with the correct one above
What is the value of 
in the equation above?
What is the value of 
To solve this problem algebraically, distribute the multiplication across each set of parentheses, remembering to multiply each term within the parentheses by its coefficient. That gives you:
Then you can combine like terms on the left side, since you have two 
terms and two numeric terms. That simplifies to:
From there, subtract 
from both sides and you have your answer, 
.
To solve this problem algebraically, distribute the multiplication across each set of parentheses, remembering to multiply each term within the parentheses by its coefficient. That gives you:
Then you can combine like terms on the left side, since you have two 
From there, subtract 
Compare your answer with the correct one above
If 
, what is the value of 
?
If 
To solve this problem, first perform algebra on the given equation to isolate the 
terms on one side of the equation and the numeric terms on the other. That means subtracting 
from both sides and adding 
to both sides to get:
You can then divide both sides by 
to realize that 
.
Now, notice that the question did not ask you for the value of 
, but rather for the value of 
. So you now need to plug in 
for 
to finish the job: 
To solve this problem, first perform algebra on the given equation to isolate the 
You can then divide both sides by 
Now, notice that the question did not ask you for the value of 
Compare your answer with the correct one above
If 
, then what is the value of 
?
If 
To answer this problem, you need to first solve for 
in the given equation. You can first multiply each side by 
to eliminate the denominator, yielding:
Then divide each side by 
to isolate 
:
Now you can plug in 
for 
in the given expression. That yields:
To answer this problem, you need to first solve for 
Then divide each side by 
Now you can plug in 
Compare your answer with the correct one above