Basic Algebra begins with variables. Anytime you have an unknown in the question, it’s a good idea to put in a variable to represent that number so you can solve using algebra.
Algebra Rules
Let’s review basic Algebra rules.
To solve a basic Algebra equation, you have to get the variable alone on one side of the equals sign, and the numbers on the other. To keep the two sides equal, anything you do to one side of the equation must be done to the other side of the equation.
To “undo” what’s being done to the variable, start by cleaning up the messiest part first. In the equation 3x - 4 = 6, first ask, “What’s being done to my variable?” In this example, it’s being multiplied by 3 and then 4 subtracted from that total. Subtracting the 4 is ‘messier’ than the other stuff because it’s hanging out there and you can’t do anything else with it. So to get rid of it, do the opposite. If 4 is being subtracted from your variable’s term, then you need to add 4 to both sides of the equation:
Solve the Equation
You should know that when you add a negative and a positive that are the same number to each other, you get zero as an answer, so our new result is:
From here, ask yourself again, “What’s being done to my variable that’s keeping it from being by itself?” Now, it’s just being multiplied by 3, so we need to do the opposite to undo that operation, that is, divide by 3 on both sides.
Any number divided by itself equals 1, and any number times 1 equals itself, so those threes cancel out to leave us with x all by itself on the left, and
10
---------
3
on the right. So
x =
10
---------
3
.
-Hint: Don’t reduce your answers, change them into decimals, or change them into mixed numbers until after you’ve finished all your work and/or looked at the answer choices. Answer choices on the SAT test are often given in improper fractions, or as an integer under a square root rather than its decimal equivalent. Save yourself work until it’s really necessary.
Your variable should always be positive at the end. If not, you will need to multiply/divide by -1 on both sides to change the signs. (It doesn’t matter whether you write the work as multiplying or dividing in this case, since the result will be the same. Just be sure to do it to ALL the numbers on both sides.)
Example Problem using PEMDAS
If one side is more complicated-looking than the other, especially if you are given parentheses, always make that one side as ‘clean’ as possible before moving numbers across the equals sign. Use PEMDAS on the SAT to get rid of your parentheses.
2(a + 3) = 10
Distribute the 2 to all terms inside the parenthesis
Then solve as normal:
If there is a fraction involved you have three ways of solving.
2p
---------
3
=4
1) You can multiply both sides by the denominator to get rid of the fraction:
3•
2p
---------
3
=4•3
2p = 12
2) You can multiply both sides by the reciprocal of the fraction attached to your variable:
3
---------
2
•
2p
---------
3
=4•
3
---------
2
p =
12
---------
2
>
3) You can cross-multiply by making the right side a fraction (putting an integer over 1):
With any of these methods, you end up with the same answer:
