Algebra logic on simplification

Question

I've been self studying algebra for some time alone and a common problem I bump into that always throws me off is something like this (Complete idiots guide to algebra, if you know a better book please tell.)

simplify 5-(-3)-(+2)+(-7)

Now the double signs are removed so the non brackets become : 5+

and brackets : -3+2-7

so now you have : 5+ -3 +2 -7

now in my example it says the equation magically becomes: 5 + 3 - 2 - 7

To me this isn't logical, How does - 3 become 3? Imagine if the -3 was money, where does the money come from? How does debt suddenly become profit? I don't even know what to call the problem I see.

I would be so grateful for someone to explain what is happening here. It is a personal hell and I can spend 3 hours looking online or ask an expert and worry about being explained to that I'm an idiot (something I'm well aware of.)

Thank you

Answer 1

You don't remove the double negative signs, you replace them with a single plus sign: $5-(-3)-(+2)+(-7)$ should reduce to $5+3-2-7$, which is $-1$. Why we should do the reduction this way isn't very intuitive, so here's an explanation should you want it.

There are two separate ideas coming together in this problem that can make it vexing.

i) Every number has an opposite:* $5$ has $-5$ (and likewise $-5$ has $5$). A number and its opposite added together make zero.**

ii) You have been lied to all your life; there is no such thing as subtraction. When we say "$5$ minus $3$" we really mean "$5$ plus the opposite of $3$", this is a mouthful, and writing $5+(-3)$ looks dumb, so it is convenient to pretend "adding the opposite of a number" is its own operation called subtraction.

So what is $5-(-3)$? Starting with (ii), we have that it's $5+(-(-3))$, or "five plus the opposite of the opposite of three." Using (i), we can say that "the opposite of the opposite of three" is three, and this leaves us with $5+(-(-3))$ being the same as $5+3$.

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*If this seems hazy, it is. The word opposite here does have a fairly specific meaning and with sufficient explication this whole point can be subsumed into (ii), but I want to avoid taking that great a detour from your main question.

**This, paired with $0+0=0$ has the odd consequence of making zero be its own opposite, which is not usually what we think of when we say opposite, but oh well.

Answer 2

Think of $5-(-3)$ as $5$ minus $-3$. The $-3$ is a loss of $3$. If you take away a loss, you're getting back what you lost. In this case what you're losing is $3$, so getting that back would mean $+3$

Say you have an bank account balance of $100$ dollars and your bank charges you a $25$ dollar fee. This would be a loss of $25$ dollars, or your balance is $100-25=75$.

Now say that bank realizes they shouldn't have charged you that $25$ dollars. They would then subtract this loss from your account by giving you back the $25$ that they took. Your balance would be $75-(-25)=75+25=100$.

The two negatives cancel out to form a positive. Here's a little chart to help you remember: $$ (+) \,\&\, (+) = (+) $$ $$ (-) \,\&\, (-) = (+) $$ $$ (+) \,\&\, (-) = (-) $$ $$ (-) \,\&\, (+) = (-) $$