If -c = 1, then why do I get c = -1 by adding two equations?

But my question is that If - c = 1, then how do we get c = -1? I am confuced. Please explain.

-c=1  Multiply both sides by -1

c = -1

Solve the following system:
{a + c = 1 | (equation 1)
-a - 2 c = 0 | (equation 2)

Add equation 1 to equation 2:
{a + c = 1 | (equation 1)
0 a - c = 1 | (equation 2)

Multiply equation 2 by -1:
{a + c = 1 | (equation 1)
0 a+c = -1 | (equation 2)

Subtract equation 2 from equation 1:
{a+0 c = 2 | (equation 1)
0 a+c = -1 | (equation 2)

Collect results:
a = 2        and          c = -1

Here is another way to look at it.

-1 * -c  =  -1 * -c        This is true since both sides are identical.

Since we know   -c  =  1   , we can plug in  1  for  -c , like this...

-1 * -c  =  -1 * 1         And simplify.

c  =  -1

Also, we can check if   c  =  -1   is a solution.

-c  =  1           Plug in  -1  for  c  and see if it makes the equation true.

-(-1)  =  1       ?

1  =  1            This is true, so   c  =  -1   is a solution.

This might seem like a lot of explanation over something simple,

but it is still good to understand I think.

If

 -c  = 1       multiply both sides by -1

(-1)(- c)   = (-1) 1

c   =    -1