I get stuck on the part where you have to add or subtract on both sides on an equation

Question

2/3n=4n-10/3n-1/2

 

Answer 1

Jesse,

 

"I get stuck on the part where you have to add or subtract on both sides on an equation"

 

It seems you are confused about a simple rule in algebra.  The fact that you can perform the same operation on both sides of an equation.

 

An equation states that the two sides represent the same number.  So 2 = 2 is an equation.  In this case you can see that both sides are the same.  If I add 4 to both sides of the equation I get 6 = 6 which is also true.

 

Suppose I write x-3 = 2.  That means that "x-3" represents the same number as 2.  If I add 3 to both sides of the equation I get x = 5.  

 

An equation says that two numbers are the same.

 

So in the case of the equation you gave, 2/3n=4n-10/3n-1/2.   Let me take a detour to discuss order of operations. If you have an expression like 2/3n or 10/3n, you need to know that the older scientific convention is that implied multiplication takes precedence over division.  This means, for example, that 2/3n means 2/(3n) and not (2/3)n.  This is the older convention where 2/3*n and 2/3n are not the same problem - and this is still used.  I think this convention is appropriate for this problem.  If one wants a different interpretation, one must use parentheses.  A couple of the solutions below ignore this older convention and end up working a different problem than the one given.  

 

This equation states that the number 2/3n is the same number as 4n - 10/3n - 1/2.   Since these two numbers are the same (except when n = 0)  I can multiply both of them by 6n and the two results will be the same.

 

So 4 = 24n² - 20 - 3n.   Now this shows that the number 4 and the number 24n² - 20 - 3n are the same number.  So I can subtract 4 from each of them and get the same result.  This means that 0 = 24n² -3n -20.

 

This gives a quadratic equation.  You can use the quadratic formula to solve it.  The solution is approximately n = .9775 or x = -.8525.    

 

You can't solve this problem by factoring since the roots of the quadratic are not rational numbers, so I assume that you know the quadratic formula.

Answer 2

The best move for an equation with fractions like this is to find the common denominator for all of the terms and then multiply both sides of the equation by that.So the four denominators are 3n, 1, 3n, and 2 therefore the common denominator is 6n. If you multiply both sides by 6n you get

6n(2/3n) = 6n( 4n - 10/3n - 1/2)  *don't forget to distribute on the right side*

6n(2/3n) = 6n(4n) - 6n(10/3n) - 6n(1/2) *simplify*

4 = 24n² - 20 - 3n *now you have no more fractions*

If you get rid of fractions and add all like terms together on either side of the equal sign knowing when to add or subtract to both sides will be much easier. In this case, since n is squared you want the equation to equal zero so the easiest thing to do is move the 4 from the left to the right. 4 is positive so do the opposite and subtract it from both sides and get

0 = 24n² - 24 - 3n *or better yet*

0 = 24n² - 3n - 24  

 

Answer 3

Hi Jesse,

If you follow a couple of simple rules of algebra, you won't have to worry about the denominators.

First, what you do to one side of the equation, you MUST do to the other side in order to keep it balanced.

Second, you need to get all of the terms that go together on one side in order to work with them.

So, in this problem, you want all of the n terms on one side and the number on the other.

2/3n = 4n -10/3n -1/2     Let's move the -1/2 first:

 +1/2 =              +1/2

2/3n +1/2 = 4n -10/3n   Now let's move the 2/3n:

-2/3n       =       -2/3n

1/2         = 4n   -12/3n    Now turn the whole number 4 into an improper fraction.

                                    If we are making 3rds, then 3/3 = 1 and 12/3 = 4

1/2         = 12/3n  - 12/3n

1/2        =  0n

1/2        =  0

This is impossible, so there is no solution.

It is OK to have different teachers show different ways to get to the answer -- as long as the rules of algebra are followed.

Answer 4

Hi,

I'm figuring your problem is really (2/3) n = 4n - (10/3)n - 1/2   ??? It's just hard to type the fraction equations in here. 

I'm looking at this problem and I see three terms that you can combine. But you have all these fractions and they will not just neatly vanish when you combine the terms. So indeed, I would do exactly what Kristopher described, multiply through by the common denominator.

The common denominator between 2/3, 10/3, and 1/2 is 6. So multiply each term by 6.

>>>This is where people mess up doing these problems. You must multiply EVERY term by 6. As Kristopher said, you must distribute the right side. Like this:

6(2/3)n = 6(4n) - 6(10/3)n - 6(1/2) 

When you multiply this out the fractions will vanish and you will get:

4n = 24n - 20n - 3

Combine like terms: 

4n = 4n - 3

Subtract 4n from both sides:

0 = 3

ooooooh! the N vanished! and 0 does NOT equal 3! This is a problem that has no answer. There is no value of N that will make 4n equal 4n - 3. So your answer is: This equation has no solution. 

That, by the way, is just fine. There are equations that have no solution. There are equations that have solutions. AND there are equations that have infinite solutions. Your teacher (if they're like me) will put all three kinds on a quiz. 

Infinite solutions? Well, if you solve an equation and get something like 5 = 5 or X = X at the end... that's an equation where any value you put in for X will work. For example:

3x - 3 = 3(x-1)      or     2n + 2(2-n) = 4    

So if the variable vanishes and you're left with a false statement, there's no answer. The answer is the null set.

If the variable vanishes and you're left with a true statement (like 4 = 4), then there are infinite answers. The answer is the set of all numbers, that is, the infinite set.