# Inverse equations

## Main Question or Discussion Point

Hello,
I have a quick question for you guys.

What is it called when you switch the signs of the equation to their opposite?

For example, does 10 + 6 = 16 invert to 10 - 6 = 4 Is this additive inverse?

Or, does 14 x 7 = 98 invert to 14 / 7 = 2 Multiplicative inverse?

lurflurf
Homework Helper
in short yes

if x*y=1 x is the multiplicatie inverse of y (also y is the multiplicatie inverse of x)
if x+y=0 x is the additive inverse of y (also y is the additive inverse of x)

Those are the nice names
bad people disrespect these numbers by calling them insults like opposite and reciprical

lurflurf
Homework Helper
Hello,
I have a quick question for you guys.

What is it called when you switch the signs of the equation to their opposite?

For example, does 10 + 6 = 16 invert to 10 - 6 = 4

Or, does 14 x 7 = 98 invert to 14 / 7 = 2
wow
no
10 + 6 = 16
becomes
-10 +(-6) = -16

14 x 7 = 98
becomes
(1/14)*(1/7)=1/98

but what you likely want is
10 + 6 = 16
so
10=16-6 or 6=16-19
and
14 x 7 = 98
so
7=98/14 or 14=98/7

This is often used with variables like
10+x=16
so
x=16-10
x=6

14 y = 98
so
y=98/14
y=7

wow
no
10 + 6 = 16
becomes
-10 +(-6) = -16

14 x 7 = 98
becomes
(1/14)*(1/7)=1/98

So, when you invert a regular equation. x + y = z the answer will always be -z

But when you invert a multiplication/division equation x * y = z the answer will always the the reciprical of z.

What is it called when you switch - to / and + to * ?

Oh, silly me, I thought this was about inverse functions. I should pay more attention.

HallsofIvy
Homework Helper
So, when you invert a regular equation. x + y = z the answer will always be -z

But when you invert a multiplication/division equation x * y = z the answer will always the the reciprical of z.

What is it called when you switch - to / and + to * ?
Perhaps it would help if you explained what in the world you are talking about! You started talking about "inverting" equations, now you are talking about inverting operations.

Yes, the "inverse" or "opposite" of addition is subtraction and the "inverse" of multiplication is division. Those apply only to specific operations, not to entire equations. I have no idea what you mean by "invert a regular equation".

If I wanted to solve x+ y= z for x, then I do the "inverse" of "add y" (since in the equation y is added to x) which is "subtract y". Subtracting y from both sides (whatever you do to one side of an equation you must do to the other) and get x+ y- y= z- y or x= z- y.

HallsofIvy, I'm curious more about inverting operations, and why:

+1 - +1 = 0 and -1 + -1 = -2 on my calculators.

HallsofIvy