Manipulating algebra, especially in your mind, can be a little demanding at first but eventually you will get so good at it that you may be even able to help your colleagues and siblings. For this topic all you have to do is memorize only two lines:
(x – y)(x + y) = x^2 – y^2…………that’s one, the other is
(x – y)^2 =/= x^2 – y^2………….(=/=) means not equal to
I wrote a topic on the “ 6 Deadly Sins to Avoid on the day of the S.A.T.” , if you take a look at example 4 , there is a question;
x^2 – y^2 = 77 and x + y = 11. What is the value of x?
You can do the long way of substituting y= 11 – x, then substituting it into the other equation so that it becomes
x^2 - (11 – x) ^2 = 77, then solve for x,
x^2 – 121 + 22x – x^2 = 77
22x = 198…………thus x =9
This was the time consuming way. Now instead of doing it this way, we are going to implement the technique of “the difference of two squares” method.
x^2 – y^2 = 77…but we know that x^2 – y^2= (x +y)(x – y)
x +y = 11 (it was given)
77 = (x + y)(x – y)……77 = 11(x – y) , thus (x – y) = 7
Now add (x + y) and (x – y) …you get 2x = 18, thus x = 9
This can be done mentally with a lot of practice, took me 20 seconds. Now let’s move on to the final topic for today, learning to save time by plugging sensible values. The question is: If half of a number is equal to 4 more than twice the number, what is the number?
a) -3
b) -8/3
c) 0
d) 7/2
e) 4
Let’s call the unknown number x. So half of the number is .5x. 4 more means 4 +, so for more twice the number means 4 + 2x. So the question in algebraic forms means
.5x = 4 + 2x ……..by just looking at the question you can tell that x is a negative value. So c), d), and e) are out. Now plug in a) – 3 and you will notice it is not the answer, hence b) is the answer. So without working out the question, it was solved by just plugging in one value, a sensible value.
This is a useful strategy when you are running low on time, but by all means take advantage of this technique to gain time. You don’t have to solve all the questions in the S.A.T.; you can use the plug in method, especially when large numbers are involved. Most people start with choice c) for plugging method, but this is not always sensible, just look at the question and choose the most obvious one to you for the easy ones and the most unobvious one for hard ones. The “plug in method” is an effective way to save time, but it is up to you to know when this method is feasible. This concludes my first lesson on algebra.
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