Let's simplify each expression step-by-step.
### Expression 1: [tex]\( 7ab - 11ab + 20ab - 31ab \)[/tex]
1. Combine like terms by adding the coefficients of [tex]\( ab \)[/tex]:
[tex]\[
(7 - 11 + 20 - 31)ab
\][/tex]
2. Calculate the coefficient:
[tex]\[
7 - 11 = -4 \\
-4 + 20 = 16 \\
16 - 31 = -15
\][/tex]
3. The simplified expression is:
[tex]\[
-15ab
\][/tex]
### Expression 2: [tex]\( 120a^2 - 345a^2 + 58a^2 - 34a^2 \)[/tex]
1. Combine like terms by adding the coefficients of [tex]\( a^2 \)[/tex]:
[tex]\[
(120 - 345 + 58 - 34)a^2
\][/tex]
2. Calculate the coefficient:
[tex]\[
120 - 345 = -225 \\
-225 + 58 = -167 \\
-167 - 34 = -201
\][/tex]
3. The simplified expression is:
[tex]\[
-201a^2
\][/tex]
### Expression 3: [tex]\( 2a + 3b - 4c + 5a + 7c - 10b \)[/tex]
1. Combine like terms for each variable separately:
[tex]\[
(2a + 5a), (3b - 10b), (-4c + 7c)
\][/tex]
2. Calculate the coefficients separately:
[tex]\[
2 + 5 = 7 \quad \text{(for \( a \))} \\
3 - 10 = -7 \quad \text{(for \( b \))} \\
-4 + 7 = 3 \quad \text{(for \( c \))}
\][/tex]
3. The simplified expression is:
[tex]\[
7a - 7b + 3c
\][/tex]
### Expression 4: [tex]\( 20x + 32x^2 - 10x + x^2 \)[/tex]
1. Combine like terms by adding the coefficients of [tex]\( x \)[/tex] and [tex]\( x^2 \)[/tex] separately:
[tex]\[
(20x - 10x), (32x^2 + x^2)
\][/tex]
2. Calculate the coefficients:
[tex]\[
20 - 10 = 10 \quad \text{(for \( x \))} \\
32 + 1 = 33 \quad \text{(for \( x^2 \))}
\][/tex]
3. The simplified expression is:
[tex]\[
33x^2 + 10x
\][/tex]
### Expression 5: [tex]\( 50xy - 34acd + 32xy - 23acd + 5 \)[/tex]
1. Combine like terms by adding the coefficients of [tex]\( xy \)[/tex] and [tex]\( acd \)[/tex] separately:
[tex]\[
(50xy + 32xy), (-34acd - 23acd)
\][/tex]
2. Calculate the coefficients:
[tex]\[
50 + 32 = 82 \quad \text{(for \( xy \))} \\
-34 - 23 = -57 \quad \text{(for \( acd \))}
\][/tex]
3. The simplified expression is:
[tex]\[
82xy - 57acd + 5
\][/tex]
### Final Results:
1. [tex]\( 7ab - 11ab + 20ab - 31ab = -15ab \)[/tex]
2. [tex]\( 120a^2 - 345a^2 + 58a^2 - 34a^2 = -201a^2 \)[/tex]
3. [tex]\( 2a + 3b - 4c + 5a + 7c - 10b = 7a - 7b + 3c \)[/tex]
4. [tex]\( 20x + 32x^2 - 10x + x^2 = 33x^2 + 10x \)[/tex]
5. [tex]\( 50xy - 34acd + 32xy - 23acd + 5 = 82xy - 57acd + 5 \)[/tex]
