Expert: Paul Klarreich - 2/8/2007

Question
Thank you so much for an explanation I can understand.  I was wondering if you could explain the reason why the -7 inside the ( ) turned to a +7 when the ( ) were taken away.

BS
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The text above is a follow-up to ...

-----Question-----
How do I go about writing an equation for this: the difference when 7 less than a number is subtracted from twice the number is 12.  And, what is the number?
-----Answer-----
Questioner:   BS
Category:  Advanced Math
Private:  no
 
Subject:  writing equations
Question:  How do I go about writing an equation for this: the difference when 7 less

than a number is subtracted from twice the number is 12.  And, what is the number?

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Hi, BS,

Sentences like this contain various mathematical elements:

A. variables, such as 'a number' or 'a second number' or 'Mary's age', or 'the distance to the plane'.  You must give them names.

Note: Many people misunderstand here.  'x' and 'm' and 'a' and 'y', are not the variables, just their names.


B. operational phrases, such as 'the difference',  'twice', 'is'.

You must translate those elements into algebraic symbols:

Variables are given symbolic names, like 'x', or 'ww', or 'a' or 'k'.
Operational phrases become the standard operational symbols, like '+', '-', '*', '='.  [In actual use, a 'star' is frequently omitted for multiplication, as we see later.]

Your sentence:  The difference when 7 less than a number is subtracted from twice the number is 12.

contains several elements:

"a number" is a variable phrase, to which we give the name "x". (Doesn't have to be this letter.)

"the number" is a variable phrase, and when we first use "a number" and then we say "the number", we are referring back, so these are the same number.

"the difference" means the result of subtraction.  [A 'sum' is the result of addition, a 'product' is the result of multiplication, etc.]

"7 less than" means the result of subtracting 7 FROM something.

"Twice" means the result of multiplying by 2.

"is" becomes "=".

We are ready to translate:

The difference when 7 less than a number is subtracted from twice the number is 12.
The difference when 7 less than    x     is subtracted from twice      x     is 12.
The difference when          x - 7     is subtracted from          2x     = 12.
The result of subtracting when x - 7     is subtracted from          2x     = 12.
         2x - (x - 7)      = 12.

That's it!

Here's a big clue:  Any time you have a phrase like 'the result of', which is part of those key words I mentioned, you probably need parentheses.

About 'And what is the number?' you just have to
A. solve the equation,
B. rewrite the sentence to see if it is correct using the solution.

2x - x + 7 = 12
x + 7 = 12
x = 5

THE NUMBER IS 5. (Always answer in a sentence.)

Now the sentence says:

The difference when 7 less than 5 is subtracted from twice 5 is 12.
The difference when (7 less than 5) is subtracted from (twice 5) is 12.
The difference when (-2) is subtracted from (10) is 12.
10 - (-2) = 12
10 + 2 = 12
RIGHT!

Sermon for the Day:  The purpose of this part of your algebra class -- applied problems -- is to develop your LANGUAGE skills, not your arithmetic skills. When you go into the real world, the ability to figure out exactly what is being said and what is not being said is a very important one.  Too bad your English teacher doesn't like to do it -- talking about Shakespeare and Dickens is more fun.  It will have to be up to your math teacher.


Answer
Questioner:   BS
Category:  Advanced Math
 
Subject:  writing equations
Question:  Thank you so much for an explanation I can understand.  I was wondering if you could explain the reason why the -7 inside the ( ) turned to a +7 when the ( ) were taken away.
....................................
Hi, BS,

I think you are referring to this sequence:
2x - (x - 7) = 12,  becomes:
2x - x + 7 = 12

You must read the symbols correctly.  The proper way to read:

- (x - 7)

is: "the opposite of the sum of  x and -7.

The opposite of a sum of two numbers is the sum of their opposites.
The opposite of x is written  -x.
The opposite of -7 is +7.

Alternatively, you can use the property:

"The result of multiplying an expression by -1 is the opposite of that expression."

Then  -(x - 7) is the same as  -1(x - 7)

Now the distributive property applies.  The factor of -1 is distributed over both terms, x and -7.

Does that do it for you?