At Westonci.ca, we connect you with experts who provide detailed answers to your most pressing questions. Start exploring now! Get immediate and reliable solutions to your questions from a knowledgeable community of professionals on our platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
Alright, let's address each part of this problem step-by-step:
### Step 1: Identifying Martina's Error
Martina attempted to subtract [tex]\((77e - 24d + 0)\)[/tex] from [tex]\((14c + 31d - 20)\)[/tex] but the result she obtained was [tex]\(0et - 3c + 7d = 20\)[/tex]. Let’s verify the subtraction manually and find her error.
Let's perform the subtraction [tex]\((14c + 31d - 20) - (77e - 24d + 0)\)[/tex]:
1. Subtract the coefficients of \'c\': there are no 'c' terms in the subtrahend.
[tex]\[ 14c \][/tex]
2. Subtract the coefficients of \'d\':
[tex]\[ 31d - (-24d) = 31d + 24d = 55d \][/tex]
3. Subtract the constant terms:
[tex]\[ -20 - 0 = -20 \][/tex]
4. There are no 'e' terms in the minuend.
So the correct difference should be:
[tex]\[ 14c + 55d - 20 \][/tex]
However, Martina got:
[tex]\[ 0et - 3c + 7d = 20 \][/tex]
Clearly, Martina made a mistake in her subtraction.
### Step 2: True/False Questions
Determine the truth value of each equation:
#### A. [tex]\(4m + 6n - 8 + (8m - 3) = 13m + 6n - 5\)[/tex]
[tex]\[ 4m + 6n - 8 + 8m - 3 = 12m + 6n - 11 \][/tex]
This does not equal [tex]\(13m + 6n - 5\)[/tex].
Answer: False
#### B. [tex]\(7.3x + 1.9y - (4.7x + 5.8y) = 2.6x - 3.9y\)[/tex]
[tex]\[ 7.3x + 1.9y - 4.7x - 5.8y = 2.6x - 3.9y \][/tex]
This statement is actually correct.
Answer: True
#### C. [tex]\(2m - \frac{1}{6}n + (m + \frac{1}{2}n) = 12m + \frac{1}{6}n\)[/tex]
[tex]\[ 2m - \frac{1}{6}n + m + \frac{1}{2}n = 3m + \frac{1}{3}n \][/tex]
Clearly, this does not equal [tex]\(12m + \frac{1}{6}n\)[/tex].
Answer: False
#### D. [tex]\(\frac{9}{4} + \frac{4}{3}(a - b) = \frac{5}{4}a - 3b\)[/tex]
Let's simplify:
[tex]\[ \frac{9}{4} + \frac{4}{3}a - \frac{4}{3}b \][/tex]
This equation does not equal [tex]\(\frac{5}{4}a - 3b\)[/tex].
Answer: False
#### E. [tex]\(2.9f + g - (1.3f - g) = 1.6f\)[/tex]
[tex]\[ 2.9f + g - 1.3f + g = 1.6f + 2g \][/tex]
The given answer is [tex]\(1.6f\)[/tex], which is not equal to [tex]\(1.6f + 2g\)[/tex].
Answer: False
### Step 3: Evaluate Expressions at [tex]\(x = -5\)[/tex]
Next, we evaluate each expression at [tex]\(x = -5\)[/tex]:
#### A. [tex]\(1.4x - 3\)[/tex]
[tex]\[ 1.4(-5) - 3 = -7 - 3 = -10 \][/tex]
#### B. [tex]\(2(x + 3) - 0.6x\)[/tex]
[tex]\[ 2(-5 + 3) - 0.6(-5) = 2(-2) + 3 = -4 + 3 = -1 \][/tex]
#### C. [tex]\(1.2x - 5 + (0.2x + 2)\)[/tex]
[tex]\[ 1.2(-5) - 5 + (0.2(-5) + 2) = -6 - 5 + (-1 + 2) = -6 - 5 + 1 = -10 \][/tex]
#### D. [tex]\(-3.5x + 2(x + 2.5)\)[/tex]
[tex]\[ -3.5(-5) + 2(-5 + 2.5) = 17.5 + 2(-2.5) = 17.5 - 5 = 12.5 \][/tex]
The expressions A and C yield -10.
### Step 4: Completing Sums and Differences
Fill in the blanks:
1. [tex]\(-9w + \_ + (4w - 2x) = -5w + 4x\)[/tex]
Here, [tex]\(-5w\)[/tex] is desired after combining like terms. Adding [tex]\(4w\)[/tex] to [tex]\(-9w\)[/tex] yields [tex]\(-5w\)[/tex]:
[tex]\[ -9w + 4w + (4w - 2x) = -5w + 4x \][/tex]
Missing term: [tex]\(4w\)[/tex].
2. [tex]\(-9w + 6x - (4w + \square) = -13w + 4x\)[/tex]
We want [tex]\(-13w\)[/tex] after adding negative terms. Adding [tex]\(-4w\)[/tex] to [tex]\(-9w\)[/tex] yields [tex]\(-13w:\)[/tex]
[tex]\[ -9w + 6x - (4w - 2x) = -13w + 4x \][/tex]
Missing term: [tex]\(-2x\)[/tex].
3. [tex]\(9w + - + (4w + 2x) = 13w - 4x\)[/tex]
We want [tex]\(13w\)[/tex] after combining terms. Adding [tex]\(4w\)[/tex] to [tex]\(9w\)[/tex] yields [tex]\(13w\)[/tex]:
[tex]\[ 9w + 4w + 2x = 13w - 4x \][/tex]
Missing term: [tex]\(4x\)[/tex].
4. [tex]\(9w - 6x - (4w + \square) = 5w - 4x\)[/tex]
We want [tex]\(5w\)[/tex] after subtracting. Adding [tex]\(-4w\)[/tex] to [tex]\(9w\)[/tex] yields [tex]\(5w\)[/tex]:
[tex]\[ 9w - 6x - (4w - x) = 5w - 4x \][/tex]
Missing term: [tex]\(2x\)[/tex].
### Step 1: Identifying Martina's Error
Martina attempted to subtract [tex]\((77e - 24d + 0)\)[/tex] from [tex]\((14c + 31d - 20)\)[/tex] but the result she obtained was [tex]\(0et - 3c + 7d = 20\)[/tex]. Let’s verify the subtraction manually and find her error.
Let's perform the subtraction [tex]\((14c + 31d - 20) - (77e - 24d + 0)\)[/tex]:
1. Subtract the coefficients of \'c\': there are no 'c' terms in the subtrahend.
[tex]\[ 14c \][/tex]
2. Subtract the coefficients of \'d\':
[tex]\[ 31d - (-24d) = 31d + 24d = 55d \][/tex]
3. Subtract the constant terms:
[tex]\[ -20 - 0 = -20 \][/tex]
4. There are no 'e' terms in the minuend.
So the correct difference should be:
[tex]\[ 14c + 55d - 20 \][/tex]
However, Martina got:
[tex]\[ 0et - 3c + 7d = 20 \][/tex]
Clearly, Martina made a mistake in her subtraction.
### Step 2: True/False Questions
Determine the truth value of each equation:
#### A. [tex]\(4m + 6n - 8 + (8m - 3) = 13m + 6n - 5\)[/tex]
[tex]\[ 4m + 6n - 8 + 8m - 3 = 12m + 6n - 11 \][/tex]
This does not equal [tex]\(13m + 6n - 5\)[/tex].
Answer: False
#### B. [tex]\(7.3x + 1.9y - (4.7x + 5.8y) = 2.6x - 3.9y\)[/tex]
[tex]\[ 7.3x + 1.9y - 4.7x - 5.8y = 2.6x - 3.9y \][/tex]
This statement is actually correct.
Answer: True
#### C. [tex]\(2m - \frac{1}{6}n + (m + \frac{1}{2}n) = 12m + \frac{1}{6}n\)[/tex]
[tex]\[ 2m - \frac{1}{6}n + m + \frac{1}{2}n = 3m + \frac{1}{3}n \][/tex]
Clearly, this does not equal [tex]\(12m + \frac{1}{6}n\)[/tex].
Answer: False
#### D. [tex]\(\frac{9}{4} + \frac{4}{3}(a - b) = \frac{5}{4}a - 3b\)[/tex]
Let's simplify:
[tex]\[ \frac{9}{4} + \frac{4}{3}a - \frac{4}{3}b \][/tex]
This equation does not equal [tex]\(\frac{5}{4}a - 3b\)[/tex].
Answer: False
#### E. [tex]\(2.9f + g - (1.3f - g) = 1.6f\)[/tex]
[tex]\[ 2.9f + g - 1.3f + g = 1.6f + 2g \][/tex]
The given answer is [tex]\(1.6f\)[/tex], which is not equal to [tex]\(1.6f + 2g\)[/tex].
Answer: False
### Step 3: Evaluate Expressions at [tex]\(x = -5\)[/tex]
Next, we evaluate each expression at [tex]\(x = -5\)[/tex]:
#### A. [tex]\(1.4x - 3\)[/tex]
[tex]\[ 1.4(-5) - 3 = -7 - 3 = -10 \][/tex]
#### B. [tex]\(2(x + 3) - 0.6x\)[/tex]
[tex]\[ 2(-5 + 3) - 0.6(-5) = 2(-2) + 3 = -4 + 3 = -1 \][/tex]
#### C. [tex]\(1.2x - 5 + (0.2x + 2)\)[/tex]
[tex]\[ 1.2(-5) - 5 + (0.2(-5) + 2) = -6 - 5 + (-1 + 2) = -6 - 5 + 1 = -10 \][/tex]
#### D. [tex]\(-3.5x + 2(x + 2.5)\)[/tex]
[tex]\[ -3.5(-5) + 2(-5 + 2.5) = 17.5 + 2(-2.5) = 17.5 - 5 = 12.5 \][/tex]
The expressions A and C yield -10.
### Step 4: Completing Sums and Differences
Fill in the blanks:
1. [tex]\(-9w + \_ + (4w - 2x) = -5w + 4x\)[/tex]
Here, [tex]\(-5w\)[/tex] is desired after combining like terms. Adding [tex]\(4w\)[/tex] to [tex]\(-9w\)[/tex] yields [tex]\(-5w\)[/tex]:
[tex]\[ -9w + 4w + (4w - 2x) = -5w + 4x \][/tex]
Missing term: [tex]\(4w\)[/tex].
2. [tex]\(-9w + 6x - (4w + \square) = -13w + 4x\)[/tex]
We want [tex]\(-13w\)[/tex] after adding negative terms. Adding [tex]\(-4w\)[/tex] to [tex]\(-9w\)[/tex] yields [tex]\(-13w:\)[/tex]
[tex]\[ -9w + 6x - (4w - 2x) = -13w + 4x \][/tex]
Missing term: [tex]\(-2x\)[/tex].
3. [tex]\(9w + - + (4w + 2x) = 13w - 4x\)[/tex]
We want [tex]\(13w\)[/tex] after combining terms. Adding [tex]\(4w\)[/tex] to [tex]\(9w\)[/tex] yields [tex]\(13w\)[/tex]:
[tex]\[ 9w + 4w + 2x = 13w - 4x \][/tex]
Missing term: [tex]\(4x\)[/tex].
4. [tex]\(9w - 6x - (4w + \square) = 5w - 4x\)[/tex]
We want [tex]\(5w\)[/tex] after subtracting. Adding [tex]\(-4w\)[/tex] to [tex]\(9w\)[/tex] yields [tex]\(5w\)[/tex]:
[tex]\[ 9w - 6x - (4w - x) = 5w - 4x \][/tex]
Missing term: [tex]\(2x\)[/tex].
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.
