Do Parentheses ALWAYS disappear after distribution?

AndromedaRXJ

This isn't a homework question. Just something I've been thinking about, so I figured I'd ask it here.

Mainly, I'm trying to find proof or disproof of whether multiplication outside of a parentheses has higher precedence than regular multiplication. Such as the difference between 3*3 and 3(3). Some people I've talked to seem to think the latter has higher precedences.

This may seem unrelated to the title, so hear me out more.

I always thought that both forms of multiplication were the same, as well as thinking that parentheses disappear after distribution. So I when I tried to prove to someone that both forms of multiplications were the same, I explained by solving:

- 2*5(x+1)=20

- 10(x+1)=20

- 10x+10=20

- 10x = 10

- x = 1

Then when plugging in.

- 2*5(1+1)=20

- 2*5(2)=20

- 10(2)=20

- 20 = 20

Then I try to show them what happens if I assume multiplication with * has lower precedence, and this was before I had any second thought about parentheses disappearing:

- 2*5(x+1)=20

- 2*5(x+1)=20

- 2*5x+5=20

- 10x+5=20

- 10x=15

- x = 1.5

Plugging in.

- 2*5(1.5+1)=20

Skipping all the steps, solving it my way, you get 25 = 20, which is just plain wrong. But THEN.... when they solved it their way, they didn't take away parentheses after distribution. So they did:

- 2*5(x+1)=20

- 2(5x+5)=20

- 10x+10=20

- 10x = 10

- x = 1

Which is basically doing the same thing as what I was doing. But I still couldn't prove that parentheses disappear after distribution. So THEN I tried putting subtraction outside of the parentheses instead of multiplication in an equation:

Their way:

- 2-5(-2x+1)=40

- 2(10x-5)=40

- 20x-10=40

- 20x=50

- x = 2.5

Plugging in:

- 2-5(-2(2.5)+1)=40

- 2-5(-5+1)=40

- 2-5(-4)=40

- 2(20)=40

- 40=40

At this point, I'm thinking "okay, maybe they're right", but I try it the other way anyway.

- 2-5(-2x+1)=40

- 2+10x-5=40

- 10x=40+5-2

- 10x=43

- x=4.3

Plugging in:

- 2-5(-2(4.3)+1)=40

- 2-5(-8.6+1)=40

- 2+43-5=40

- 45-5=40

- 40=40

At this point, I just lose it. After all that, both methods seem to work. But there's a consequence! I have two equations saying:

2-5(-2(2.5)+1)=40

and

2-5(-2(4.3)+1)=40

Which is nonsensical!

I tried some online calculators which ultimately said the way I initially thought was correct, but that doesn't actually tell me anything. I actually want to see a demonstration of why either one is correct or wrong.

If one method is wrong, I want to see a mathematical consequence that shows it. Basically, I want to see one of the methods fail.

Iseous

2 - 5*(-2(2.5) +1) = 22, not 40.
You multiplied 2*[-5(-2(2.5) + 1)] rather than add:
2 + [-5(-2(2.5) + 1)] = 22

In the other case, you added, which was correct. In an equation with one variable raised to the first power, you will only have one solution, not two.

As far as parentheses, they take precedence because you are looking at the whole thing. For your example:
2 - 5(-2x + 1) = 40

Take the parenthesis part and assume it is a variable, since you want the whole thing. So say y = (-2x + 1)
Thus,
2 - 5y = 40

solve for y:
y = -38/5 = -7.6

Then go back to your parenthesis:
-7.6 = (-2x + 1)
-2x = -8.6
x = 4.3

As for the first example, you can multiply those in any order:
2*5(x+1)=20

Again, you can separate the parenthesis into a variable to see it clearer:
y = (x +1)
2*5*y = 20

In multiplication, you can multiply these in any order and it will give you the same answer, so you do not have to get rid of or keep the parenthesis part right away.

I'm not sure if that helped explain it at all.

AndromedaRXJ

I know things INSIDE parenthesis take precedence. It's things directly outside of parenthesis that I'm asking about.

Some people I talk to think that in:

2*5(x+1)=20

You distribute the 5 to the parenthesis before multiplying the 2*5.

Iseous

Sorry, I just edited it. For that example, you can distribute the 5 first or multiply the 2 and the 5 first; it won't matter for the solution. Although you have to keep the parenthesis if you are distributing the 5 since the 2 must be multiplied by that whole part in the parenthesis.

AndromedaRXJ

So basically, parenthesis don't always have to disappear after distribution?

And all forms of multiplication have equal precedence?

Since that's basically what I ultimately want to know.

Iseous

Well, actually, they shouldn't disappear after distribution if you want to multiply by the parenthesis. For instance:
2 * 5*(3 - 2)
Distribute the 5:
2 * (15 -10)
2 * (5) = 10

If you took them away:
2 * 5*(3 - 2)
2 * 15 - 10
30 - 10 = 20

If you take away the parenthesis, you would not be multiplying the 2 by the (-10). If you were to say y = (5 - 3), then you have:
2*5*y

You want to multiply the 2 and the 5 by the entire y, not just the 5 or -3 part.

For multiplication, yes, you can multiply things in any order. If you have parenthesis, then try to treat it as a variable if that helps. So, for example:
2*5*y -> You can interchange those values in any way you want such as 5*y*2 and you would get the same thing as long as you apply it to the entire expression in the parenthesis.

AndromedaRXJ

Well thanks for giving me all your input on this. Had to go and mess around with it all again, but it really helped out a lot. :)

Iseous

You're welcome. I'm glad I was able to help.

Mute

You shouldn't write your equations with the "-" in front of them like that. I thought that they were minus signs and everything was probably going wrong because you weren't distributing them properly! You should use an arrow "->" or something that doesn't look like a mathematical operation if you really want to indicate some sort of flow from one line to the next. The space between the dash and the equations isn't enough to distinguish from minus signs, I think.

micromass

Yes!! This confused me so much at first.

Please never use "-" to start a new line.

AndromedaRXJ

Oh, sorry about that >_<