a-[b-c+a-{b-(c-a-a+b)}]
First we look for the innermost grouping symbols, that is,
the grouping symbols which have no grouping symbols within them,
I will color them red:
a-[b-c+a-{b-(c-a-a+b)}]
We combine the "-a-a" by considering it as "-1a-1a" to get "-2a",
and we replace "-a-a" by "-2a" and copy everything else over:
a-[b-c+a-{b-(c-2a+b)}]
Next we remove those innermost grouping symbols by the rule that
since it is preceded by a "-" sign we drop the grouping symbols by
changing all the signs before all the terms. So we replace
"-(c-2a+b)" by "-c+2a-b" and copy everything else over:
a-[b-c+a-{b-c+2a-b}]
Next we look for the innermost grouping symbols again, that is,
the grouping symbols which have no grouping symbols within them,
I will color them red:
a-[b-c+a-{b-c+2a-b}]
We see within the { } that thee "b" and the "-b" add to zero, so
we just eliminate them and copy everything else over:
a-[b-c+a-{-c+2a}]
There are no like terms to combine within those grouping symbols,
so we remove those grouping symbols again by the rule that since
it is preceded by a "-" sign we drop the grouping symbols by
changing all the signs before all the terms. So we replace
"-{-c+2a}" by "+c-2a" and copy everything else over:
a-[b-c+a+c-2a]
We see within the [ ] that the "-c" and the "+c" add to zero, so
we just eliminate them and copy everything else over:
a-[b+a-2a]
We combine the "+a-2a" by considering it as "+1a-2a" to get "-1a",
or just "-a" and we replace "+a-2a" by "-a" and copy everything else
over:
a-[b-a]
There are no like terms to combine within those final grouping
symbols, so we remove those final grouping symbols again by the rule
that since it is preceded by a "-" sign we drop the grouping symbols
by changing all the signs before all the terms. So we replace
"-[b-a]" by "-b+a" and copy everything else over:
a-b+a
We combine the "a" and "+a" by considering them as "1a" and "1a" to get
"2a" and we replace them by by "2a" and copy everything else
over, winding up with
2a-b
Edwin
