When Is Subtraction Associative?

Algebra Level 1

$\large (a-b) - c = a - (b - c)$

Suppose $a, b$ and $c$ are numeric variables. Under what conditions is the equation above true?

Always true Only true when at least one of the variables is equal to 0 Only true when

c=0

Only true when all of the variables are equal to 0 Never true

2 solutions

Damon Demas Staff
Mar 2, 2016

In an expression without parentheses, such as $a - b - c,$ the subtraction on the left is performed first. So the given equation $(a - b) - c = a - (b - c)$ is equivalent to $a - b - c = a - (b - c).$ This can be simplified by distributing the subtraction on the right side: $a - b - c = a - b + c.$ Subtracting $a$ from both sides yields $- b - c = - b + c,$ and adding $b$ to both sides of this gives $- c = c.$ Finally, subtracting $c$ from both sides yields $-2c = 0,$ which is equivalent to $c = 0$ . So the original equation is only true when $c = 0$ .

but if all vars are = 0, both sides equate to 0, 0=0

Adrian Craven - 4 years, 11 months ago

Good point! The word 'only' in the answer choices makes the difference here - since the situation in which all the variables are equal to 0 isn't the only way the equation can be true (it's also true when c =0 and a and b are nonzero), the answer 'only true when all of the variables are equal to 0' turns out to not be the final answer.

Damon Demas Staff - 4 years, 11 months ago

saying c=0 is the only solution is false as (a=0, b=0 and c=0) is also a solution. As( a=0, b=0 and c=0) is not the only solution, I think the best answer is "at least one variable is 0 and this variable must be c".

Gerard Boileau - 3 years, 4 months ago

@Gerard Boileau – The answer is not saying that c=0 is the one and only solution, since this tells us nothing about a or b. Rather, it is saying that there are multiple solutions, all of which share the property that c=0. Since this is true of a,b,c=0, it is a solution, but not the only solution.

Jason Carrier - 2 years, 10 months ago

Being a professional where math isn't something I routinely come across, this one threw me for a loop! Then seeing the proof above, seeing the 3rd step right side change to =a-b+c, I couldn't get my brain around it for a couple of minutes. Then it clicked. Now, I just hope it sticks ;)

Jason Cox - 3 years, 3 months ago

Gia Hoàng Phạm
Sep 22, 2018

$(a-b)-c=a-(b-c)=a-b-c=a-b+c \implies (a-b-c)-(a-b+c)=0 \implies a-b-c-a+b-c=0 \implies -2c=0$ if & only $c=\boxed{\large{0}}$

When Is Subtraction Associative?

2 solutions

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