Solving Equations
At its core, algebra is like placing your coat on the movie seat next to you while your friend goes to get popcorn. People looking at that seat don’t actually see your friend. What they see is a placeholder meant to signify that a person is sitting there. If we were to make an equation to represent this, it would be:
coat = your friend
or simply:
c = friend
That’s what algebra is all about. Letters (variables) are used to represent an undefined quantity in an expression or equation. So in the equation 3w + 6 = 23, the variable w is merely a placeholder for some real number that has yet to be determined.
Typically, you need to get to the bottom of the mystery and determine the numeric value of w. This is called manipulating the equation by isolating the variable. It sounds a bit wicked, but you won’t get in trouble for doing it. Isolating the variable means you want to mess with the equation until you are left with only the variable w on one side of the equation. An actual number should be on the other side of the equals sign. To manipulate the equation, you can add, subtract, multiply, divide, square, or take the square root of numbers, but whatever you do to one side of the equation, you must also do the other side. You can be manipulative, but you have to be fair about it.
The starting point:
3w + 6 = 23
subtracting 6 from both sides:
3w + 6 – 6 = 23 – 6
3w = 17
dividing 3 from both sides:
Ta da! The variable has been isolated, and by manipulating the equation you get an actual numeric value for it.
A majority of the algebra items you see on the SAT will require some bit of manipulation. We can bring back our sample item from the Anatomy section and manipulate it to find the answer.
4. If and , what does b equal?
Start with the equation. Here we want to isolate b:
Now we plug in the given value for a.
Then we square both sides.
4096 = b
This item brings up a good point. The equation with the variable w is pretty straightforward. Most real SAT items aren’t. They are filled with fractions, exponents, and k, where k is a variable representing the kitchen sink. It’s all still manipulation, though, so remember to do to one side what you do to the other, and it will all work out.
Distribution and Factoring
These are two nifty little equation-manipulating gimmicks that crop up on algebra items from time to time. Distribution takes a term outside a set of parentheses and distributes it across all the terms inside the parentheses. So if you have 6(3w + 8), you could distribute the 6 in the following manner:
6(3w + 8) = (6)(3w) + (6)(8) = 18w + 48
“Great, but so what?” you might say in your most bored, underwhelmed voice. Well, it just so happens that many really complicated-looking algebra items become simpler after distribution. But don’t take our word for it:
By distributing, you now have two similar terms that cancel each other out:
Factoring is distribution in reverse. With factoring, you notice common factors that can be taken out of an equation. Starting with , you can take out the greatest common factor, 7rb, so that the factored equation becomes .
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