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Sagot :
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]
### 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]
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