Question 8:

Martina subtracted [tex]$(77e + 24d + 0)$[/tex] from [tex]$(14c + 31d - 20)$[/tex] to get [tex]$0e + t - 3c + 7d = 20$[/tex]. She realized she made an error when she subtracted. What error did Martina make? Show how to find the correct difference.

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Select True or False for each equation:

A. [tex]$4m + 6n - 8 + (8m - 3) = 13m + 6n - 5$[/tex]
- True
- False

B. [tex]$7.3x + 1.9y - (4.7x + 5.8y) = 2.6x - 3.9y$[/tex]
- True
- False

C. [tex]$2m - \frac{1}{6}n + \left(m + \frac{1}{2}n\right) = 12m + \frac{1}{6}n$[/tex]
- True
- False

D. [tex]$\frac{9}{4} + \frac{4}{3}(a - b) = \frac{5}{4}a - 3b$[/tex]
- True
- False

E. [tex]$2.9f + g - (1.3f - g) = 1.6f$[/tex]
- True
- False

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Find the value of each expression when [tex]$x = -5$[/tex]. Circle the expressions that have a value of -10 to show which expressions are equivalent to one another:

A. [tex]$1.4x - 3$[/tex]

B. [tex]$2(x + 3) - 0.6x$[/tex]

C. [tex]$1.2x - 5 + (0.2x + 2)$[/tex]

D. [tex]$-3.5x + 2(x + 2.5)$[/tex]

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Use terms from the box to complete the sum or difference:

[tex]\[
\begin{array}{l}
-9w + \_ + (4w - 2x) = -5w + 4x \\
-9w + 6x - (4w + \square) = -13w + 4x \\
9w + \_ + (4w + 2x) = 13w - 4x \\
9w - 6x - (4w + \square) = 5w - 4x
\end{array}
\][/tex]

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].