Standardized tests try to avoid having students “solve for x.” Asking a question this simples leaves students the opportunity to plug in the answer choices (A – E) until they find one that works. This doesn’t necessarily test whether a student knows algebra but can instead test whether a student knows how to take a multiple-choice test. To avoid this, the SAT often asks students to solve for a statement instead of a singular variable. Let’s look at an example:

1. If 6x + 4 = 7, what is the value of 6x – 4?
A. –7
B. -1
C. 1
D. 7
E. 8
Notice that the problem is asking for the value of “6x – 4,” a mathematical statement, instead of asking for the student to solve for “x,” a variable. You have several options on how to approach this problem. Most students approach the problem like this:

Solve for x:
6x + 4 = 7
x = ½
Plug x into the statement:
6x – 4 = 6 (1/2) – 4 = 3 – 4 = -1
This is valid, but there is a much faster way (and we’ll see in a moment that the previous method does not always work).

Instead, ask yourself, “How can I create 6x – 4 in this equation?” Well, 6x + 4 already looks very similar to what we want to solve for. The only difference is a positive 4 instead of a negative 4. To change 6x + 4 into 6x – 4, all we need to do is subtract 8.
6x + 4 = 7
-8 and -8
6x – 4 = -1
Both methods result in the answer of -1.

Here is a similar example:
2. If (2x – 5)(2x + 5) = 5, what is the value of 4x^2?
A. -30
B. -20
C. 10
D. 20
E. 30
*The answer is provided at the end of the post.

Now, let’s look at an example that cannot be solved by the usual student method of solving for the variable.
3. If a + 2b is equal to 125 percent of 4b, what is the value of a/b?
This is a free response question, so there are no answer choices. Notice that the problem gives the student one equation but two variables. This equation cannot be solved for the variables separately. In this case, we must create a/b in the equation.

First, let’s create the equation:
a + 2b = 1.25 (4b)
Now, we must create the statement a/b:
a + 2b = 5b
a = 3b
a/b = 3
The correct answer is 3.

Here is a similar example:
4. √(x^2-t^2 )=2t-x
If x and t are positive numbers that satisfy the equation above, what is the value of x/t?
*The answer is provided at the end of the post.

Answers: #2 E
#4 5/4

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