Simplification help

[snakehead]

Why can I not simplify this equation like this [1]:

x + 5/2 (x) + 7/2 (x) = 45.5
6 (3x) = 45.5

I don't understand why that's wrong, and this is correct [2]:

x + 2.5 (x) + 3.5 (x) = 45.5
7x = 45.5

(I understand how [2] is achieved but why can I not in [1] just group up the like terms; add the Xs and then the integers and then multiply?)

*I understand that the fractions and decimals are equivalent. [2] makes sense but what is wrong with the 1st method.

Thanks (sorry if this is confusing)

 

[lev888]

Why can I not simplify this equation like this [1]:

x + 5/2 (x) + 7/2 (x) = 45.5
6 (3x) = 45.5

I don't understand why that's wrong, and this is correct [2]:

x + 2.5 (x) + 3.5 (x) = 45.5
7x = 45.5

(I understand how [2] is achieved but why can I not in [1] just group up the like terms; add the Xs and then the integers and then multiply?)

*I understand that the fractions and decimals are equivalent. [2] makes sense but what is wrong with the 1st method.

Thanks (sorry if this is confusing)

Please add more steps to your solution, it's not clear what you are doing (where do 6 and 3x come from?).

 

[Dr.Peterson]

Why can I not simplify this equation like this [1]:

x + 5/2 (x) + 7/2 (x) = 45.5
6 (3x) = 45.5

I don't understand why that's wrong, and this is correct [2]:

x + 2.5 (x) + 3.5 (x) = 45.5
7x = 45.5

(I understand how [2] is achieved but why can I not in [1] just group up the like terms; add the Xs and then the integers and then multiply?)

*I understand that the fractions and decimals are equivalent. [2] makes sense but what is wrong with the 1st method.

Thanks (sorry if this is confusing)

It isn't clear exactly what you are doing to get your result.

The correct work, with details filled in, is like this:

x + (5/2)x + (7/2)x = 1x + (5/2)x + (7/2)x = (1 + 5/2 + 7/2)x = (2/2 + 5/2 + 7/2)x = ((2+5+7)/2)x = (14/2)x = 7x​

Here I grouped like terms, adding their coefficients while factoring out the x. I used the distributive property to factor out the x, and appropriate properties of fractions to do the addition.

You appear to be doing something like this, which is wrong at several points:

x + (5/2)x + (7/2)x = x + (5/2 + 7/2)(x + x) = x + 6(x+x) = 6(x+x+x) = 6(3x) = 18x​

Why do you think that is right?

I suspect you may have made two mistakes: adding the fractions incorrectly, and adding the x's rather than factoring out a single x.

Whenever you do something in math, you must have a reason for doing it; it's not that you can do whatever you want and the burden is on others to show you are wrong!

 

[snakehead]

x + (5/2)x + (7/2)x = x + (5/2 + 7/2)(x + x) = x + 6(x+x) = 6(x+x+x) = 6(3x) = 18x

This is what I am doing. I don't understand why you can't just group the fractions (5/2 + 7/2) , add those, group the Xs (x + x + x) and then add those. This is my reasoning.

What should I look into to amend this mistake?

+ (5/2)x + (7/2)x = 1x + (5/2)x + (7/2)x = (1 + 5/2 + 7/2)x = (2/2 + 5/2 + 7/2)x = ((2+5+7)/2)x = (14/2)x = 7x

I understand what you've done here. It's not that I don't accept it (I know it is correct), I just don't understand why you can't add the Xs.

*Also is it OK to just multiply 1/1 by 2 to make it an equivalent fraction (2/2) and leave 5/2 and 7/2 as they are (I know 1/1 and 2/2 are the same thing, I just feel uneasy about only multiplying one fraction for some reason)?

 

[Deleted member 4993]

Suppose, you had:

5 shirts + 7 shirts = how many shirts do you have?

In "your way" of calculating:

5 shirts + 7 shirts = (5 + 7) (shirts + shirts) = 12 (2 shirts) = 24 shirts .............................. This is incorrect (as you can count those out)

 

[topsquark]

This is what I am doing. I don't understand why you can't just group the fractions (5/2 + 7/2) , add those, group the Xs (x + x + x) and then add those. This is my reasoning.

What should I look into to amend this mistake?


I understand what you've done here. It's not that I don't accept it (I know it is correct), I just don't understand why you can't add the Xs.

*Also is it OK to just multiply 1/1 by 2 to make it an equivalent fraction (2/2) and leave 5/2 and 7/2 as they are (I know 1/1 and 2/2 are the same thing, I just feel uneasy about only multiplying one fraction for some reason)?

Factor! Let's be a bit simpler and add x + 2x. You would have (1 + 2)(x + x) = (3)(2x) = 6x. Clearly wrong. But if you factor the common x out on the right:
x + 2x = (1 + 2)x.

-Dan

 

[lev888]

Why can I not simplify this equation like this [1]:

x + 5/2 (x) + 7/2 (x) = 45.5
6 (3x) = 45.5

I don't understand why that's wrong, and this is correct [2]:

x + 2.5 (x) + 3.5 (x) = 45.5
7x = 45.5

(I understand how [2] is achieved but why can I not in [1] just group up the like terms; add the Xs and then the integers and then multiply?)

*I understand that the fractions and decimals are equivalent. [2] makes sense but what is wrong with the 1st method.

Thanks (sorry if this is confusing)

When we say "grouping like terms" we mean use the commutative property of addition: a+b=b+a
This allows us to move terms around in order to group like terms together: 2x + 3 + 5x = 2x + 5x + 3.
Then we can apply the distributive property of addition and multiplication: 2x + 5x + 3 = (2+5)x + 3
Then we add: (2+5)x + 3 = 7x + 3.

You can't invent your own rules. You can propose whatever new operatios you want ("add the Xs and then the integers"), but you still have to justify them using the existing math laws.

 

[Dr.Peterson]

Also is it OK to just multiply 1/1 by 2 to make it an equivalent fraction (2/2) and leave 5/2 and 7/2 as they are (I know 1/1 and 2/2 are the same thing, I just feel uneasy about only multiplying one fraction for some reason)?

You aren't multiplying 1/1 by 2; you're multiplying its numerator and denominator both by 2, which leaves its value unchanged. This is just rewriting the fraction, not changing it, so you can do that any time you want, to any individual fraction.

 

[JeffM]

x + 2.5x + 3.5x = 1x + 6x = 7x.

You forgot that x means 1x.

I remember making the same mistake when I was 12 years old. Felt like an idiot. Still remember it 65 years later.

 

[snakehead]

You aren't multiplying 1/1 by 2; you're multiplying its numerator and denominator both by 2, which leaves its value unchanged. This is just rewriting the fraction, not changing it, so you can do that any time you want, to any individual fraction.

Thank you

x + 2.5x + 3.5x = 1x + 6x = 7x.

You forgot that x means 1x.

I remember making the same mistake when I was 12 years old. Felt like an idiot. Still remember it 65 years later.

Yeah, it's this that I forget, that I can make x = (1/1)x. Thanks ^

Thank you all very much for your help!