Solve and check the equation for the indicated variable, 2(n-7)+3=9

Question

How do I find the indicated variable (n) for this and other problems. If I only knew an easy method to do these...it would be fantastic!

 

Another example:

 

8e+3(5-e)=10

 

Please explain step by step!

Answer 1

Sometimes you can take different approaches, and it can depend on what you've got.  In fact, your two examples would be worked differently so it can be difficult to just memorize a series of steps.
 
The thing to remember is that you have to isolate to your variable.  In your first equation, 2(n - 7)+3=9, that's a little easier to do.
 
Let's look at what you've got.  Our variable, n, is inside parenthesis.  You do have to know order of operations, but you can't always do the actual calculations when you have variables in the mix, but you do have to understand it so you know what you can and cannot do.  You cannot solve what's inside the parenthesis, but you can keep it together "as a set."  That is, if you did parenthesis first, you would then multiply the 2 by those results.  But you can't do the n-7, so for now it just has to stay inside the parenthesis.
 
Now, you can distribute the 2, which would eliminate the parenthesis issue, but you don't want to.  You have to divide both the 3 and 9 by 2 as well, which will end up in fractions and just make the thing worse.  (By the way, I don't consider it "bad" if you do something like that.  You'll soon figure out it wasn't the best approach, and hopefully learn something from the process. ;-) )
 
But look what else you have... that 3 sitting there as a term all by itself.  Since we need to eliminate everything else and get it down to just n, and since it's "legal" to subtract the 3 off, that's a good thing to do.  Even though that addition would come last in order, we'd be adding it to the result of 2(n-7), so as long as we keep that 2(n-7) intact, it's OK.  The n isn't in that term with the 3, so it's good to just get rid of the term.  That's a step closer to isolating n.
 
Remember you have to do the same thing to both sides of the equation in order to make it still equivalent to what you have, so we must subtract 3 from both sides:
 
2(n-7) + 3 = 9
          - 3        -3
 
2(n-7) = 6
 
Always remember to do the opposite operation as what is there.  The 3 is an add, a plus, so we do a minus to get rid of it, and do it to both sides.
 
Still going towards that idea of isolating n.... we still have to keep the (n-7) as a set for the moment.  That is, we can't get rid of the 7 for now, because it's inside the parenthesis and we can't do that subtraction since we don't know what n is.  (By the way, it's still legal to distribute the 2, but you'd just have to factor it back out, so there's no point.  You're trying to isolate to n, not put more on it.)
 
So how about we eliminate what's outside the parenthesis?  Since that's multiplication, we would divide it.  That's easier to do now, because we got rid of the 3 and we won't end up with fractions.  So we divide both sides by the 2:
2(n-7) = 6
-------     ----
  2           2
 
On the left side, the 2 "cancels out."  2/2 = 1, so that makes it 1(n-7), and 1 times anything is itself.  So that essentially just eliminates the 2 on that side.  The right side is now a nice neat 3 instead of a fraction.  Giving us:
n-7 = 3
 
Now our parenthesis can go away.  The reason is the 2 isn't there anymore.  It's no longer 2 times "the result of" n - 7.  It's now just n - 7.
 
So at this point we will eliminate the 7 in order to isolate to just the n.  Since that's a minus, we add 7 to both sides:
n-7 = 3
+7      +7
 
n = 10
 
Check:
2 (10 - 7) + 3 = 9
2 (3) + 3 = 9
6 + 3 = 9
9 = 9

 

If there are rules in there you don't understand, ask.  Best that you learn it now. ;-)