Which is this? Negation or subtraction?

Question

I feel like apologizing for asking because the answer to this debate seems obvious to me, but it seems equally but oppositely obvious to the other person ... and that would mean apologizing doubly if I'm wrong. (Still prepared to do so, but if I need to issue two apologies I'd rather do them at the same time.) I'll try to phrase this so that you cannot tell which is my interpretation.

Here's the equation in question:

34 - 6 x 5 = 4

Here are the two descriptions of how the left side of that equation should be simplified (and I flipped a coin to determine which should go first).

Interpretation A

1. 6 x 5 must be resolved first, making the expression 34 - 30
2. 34 - 30 can then be simplified to 4
3. It must be done this way because it's a trinomial with terms of 34, 6, and 5 with a subtraction operator separating the first and the second terms and a multiplication operator separating the second and third terms. Because the second term is involved in both operations, the multiplication takes precedence before anything else can be done with that term.

Interpretation B

1. Since -6 is the second term in the polynomial, you can apply the concept of the "additive inverse" here and restate 34 - 6 x 5 as 34 + (-6) x 5
2. 34 + (-6) x 5 can be simplified to 34 + -30
3. That can be simplified to 4

Now, I am trying to present both interpretations in the best light as if both people are here to defend their view so I apologize if I don't capture the other person's logic perfectly.

The proponent of Interpretation A says their solution is the simplest, strictest application of the order of operations and any other measures are unnecessary. They also say that by calling the second term (-6) that person is introducing an operation (+) that is not in the original statement.

The proponent of Interpretation B states that given the definition of the additive inverse :: a - b = a + (-b) (they provided an external reference for what they used from a valid source that we both accept, in case the context here is oversimplifying things) :: that it is valid to rewrite the left hand expression as they did. Furthermore, because of additive inverses there is no distinction between subtracting 6 and adding -6.

Getting back to that "missing operator": The proponent of A said that if that minus sign between the first and second terms were to be interpreted as a unary negation operator instead of a binary subtraction operator, then

34 - 6 x 5 should have been written as 34 (-6) x 5

and that gap between the first two terms implies multiplication, not addition, resulting in a simplified value of -900

So,

• are both interpretations valid?
• are both interpretations wrong?
• if only one is valid, which is it?

And to be fair, if they are both valid then the issue of which is better here isn't really relevant to the question at hand (strictly speaking).

Edit: I felt I should add this as well. As is often the case, this debate was spawned by a meme and here it is ... perhaps you see something relevant one or both of us missed: soccer/football player math meme

Answer 1

You are not taking into account the precedence of operations. "PEDMAS" (or, for ourI British friends, "BODMAS"). "Parentheses, Exponents, Division, Multiplication, Addition, Subtraction". (Or, British, "Brackets, Orders, Division, Multiplication, Addition, Subtraction".)

34- 6x5 we MUST do the multiplication first. IF 34- 6 is intended first, use parenthese: (34-6)x 5= 28x5= 140.