Let's solve each part step by step:
### Part (a)
Simplify the expression: [tex]\( 4m + ((6m - 3n) - (9n - 5m) + (8m - 2n)) \)[/tex]
1. Distribute and simplify within the inner parentheses:
[tex]\[
(6m - 3n) - (9n - 5m) + (8m - 2n)
\][/tex]
Rewrite subtraction as addition of negative terms:
[tex]\[
= (6m - 3n) + (-9n + 5m) + (8m - 2n)
\][/tex]
2. Combine the like terms:
[tex]\[
= 6m + 5m + 8m - 3n - 9n - 2n
\][/tex]
Add the 'm' terms together and the 'n' terms together:
[tex]\[
= (6m + 5m + 8m) + (-3n - 9n - 2n)
\][/tex]
[tex]\[
= 19m - 14n
\][/tex]
3. Now include the term from outside the parentheses:
[tex]\[
4m + 19m - 14n
\][/tex]
Combine the 'm' terms:
[tex]\[
23m - 14n
\][/tex]
The simplified expression is:
[tex]\[ 23m - 14n \][/tex]
### Part (b)
Simplify the expression: [tex]\( 2a - \{7a - (3a - 7b) + (10a - 9b)\} \)[/tex]
1. Simplify the expression within the curly braces:
[tex]\[
7a - (3a - 7b) + (10a - 9b)
\][/tex]
Distribute and simplify inside the parentheses:
[tex]\[
= 7a - 3a + 7b + 10a - 9b
\][/tex]
Combine the like terms:
[tex]\[
= (7a - 3a + 10a) + (7b - 9b)
\][/tex]
[tex]\[
= 14a - 2b
\][/tex]
2. Now include the term from outside the curly braces:
[tex]\[
2a - (14a - 2b)
\][/tex]
Distribute the negative sign:
[tex]\[
= 2a - 14a + 2b
\][/tex]
Combine the 'a' terms:
[tex]\[
= -12a + 2b
\][/tex]
The simplified expression is:
[tex]\[ -12a + 2b \][/tex]
### Part (c)
Simplify the expression: [tex]\[ -(-6x + 3y - (8x - [2y - 4x - 26xy + 10x] - 9y) + 12x) \][/tex]
1. Simplify within the square brackets first:
[tex]\[
2y - 4x - 26xy + 10x
\][/tex]
Combine the 'x' terms:
[tex]\[
= 2y + (10x - 4x) - 26xy
\][/tex]
[tex]\[
= 2y + 6x - 26xy
\][/tex]
2. Substitute back into the inner parentheses:
[tex]\[
-6x + 3y - (8x - (2y + 6x - 26xy) - 9y) + 12x
\][/tex]
Distribute and simplify within the inner parentheses:
[tex]\[
= -6x + 3y - (8x - 2y - 6x + 26xy - 9y) + 12x
\][/tex]
[tex]\[
= -6x + 3y - (2x - 11y + 26xy) + 12x
\][/tex]
Distribute the negative sign:
[tex]\[
= -6x + 3y - 2x + 11y - 26xy + 12x
\][/tex]
Combine the like terms:
[tex]\[
= (-6x - 2x + 12x) + (3y + 11y) - 26xy
\][/tex]
[tex]\[
= 4x + 14y - 26xy
\][/tex]
3. Distribute the outer negative sign:
[tex]\[
= -(4x + 14y - 26xy)
\][/tex]
[tex]\[
= -4x - 14y + 26xy
\][/tex]
The simplified expression is:
[tex]\[ -4x - 14y + 26xy \][/tex]
