Problem Analysis:
The task involves solving a series of algebraic equations to find the value of \( x \) for each equation. Once the values of \( x \) are determined, they will be used to decode a message by matching them with the corresponding letters provided in the answer key.
Step-by-Step Solution:
#### 1. Solve Each Equation for \( x \):
Equation 1: \( 1 \cdot x = 10 \cdot 8 \)
\[
x = 10 \cdot 8 = 80
\]
Equation 2: \( 2 \cdot x = 36 \)
\[
x = \frac{36}{2} = 18
\]
Equation 3: \( 4 \cdot x = 56 \)
\[
x = \frac{56}{4} = 14
\]
Equation 4: \( x + 6 = 3 \)
\[
x = 3 - 6 = -3
\]
Equation 5: \( x - 17 = -23 \)
\[
x = -23 + 17 = -6
\]
Equation 6: \( x + 9 = 34 \)
\[
x = 34 - 9 = 25
\]
Equation 7: \( 5 \cdot x = 115 \)
\[
x = \frac{115}{5} = 23
\]
Equation 8: \( 2 \cdot x = 13 \)
\[
x = \frac{13}{2} = 6.5
\]
Equation 9: \( x + 5 = 7 \)
\[
x = 7 - 5 = 2
\]
Equation 10: \( 10 \cdot x = 52 \)
\[
x = \frac{52}{10} = 5.2
\]
Equation 11: \( 11 \cdot x = 42 \)
\[
x = \frac{42}{11} \approx 3.818
\]
Equation 12: \( 4 \cdot x = 96 \)
\[
x = \frac{96}{4} = 24
\]
Equation 13: \( x + 49 = -54 \)
\[
x = -54 - 49 = -103
\]
Equation 14: \( 1 \cdot x + 9 = 8 \)
\[
x = 8 - 9 = -1
\]
Equation 15: \( x + 13 = 5 \)
\[
x = 5 - 13 = -8
\]
Equation 16: \( 3 \cdot x = 37 + 16 \)
\[
3 \cdot x = 53 \implies x = \frac{53}{3} \approx 17.667
\]
Equation 17: \( x + 27 = 27 \)
\[
x = 27 - 27 = 0
\]
2. Match Solutions to the Answer Key:
The solutions for \( x \) are:
\[
80, 18, 14, -3, -6, 25, 23, 6.5, 2, 5.2, 3.818, 24, -103, -1, -8, 17.667, 0
\]
Using the provided answer key:
\[
\begin{aligned}
&\text{a. } 72 & &\text{b. } 2 \\
&\text{c. } 14 & &\text{d. } -5 \\
&\text{e. } -39 & &\text{f. } 12 \\
&\text{g. } 35 & &\text{h. } 18 \\
&\text{i. } -18 & &\text{j. } 23 \\
&\text{k. } 23 & &\text{l. } 25 \\
&\text{m. } 26 & &\text{n. } -6 \\
&\text{o. } 41 & &\text{p. } -2 \\
&\text{q. } -54 & &\text{r. } -6 \\
&\text{s. } -6 & &\text{t. } 65 \\
&\text{u. } 12 & &\text{v. } 33 \\
\end{aligned}
\]
Matching the solutions to the closest values in the answer key:
\[
\begin{aligned}
&80 \to \text{No match (closest is 72, but not exact)} \\
&18 \to \text{h. } 18 \\
&14 \to \text{c. } 14 \\
&-3 \to \text{d. } -5 \quad (\text{Closest}) \\
&-6 \to \text{n. } -6 \\
&25 \to \text{l. } 25 \\
&23 \to \text{j. } 23 \quad \text{or} \quad \text{k. } 23 \\
&6.5 \to \text{No exact match} \\
&2 \to \text{b. } 2 \\
&5.2 \to \text{No exact match} \\
&3.818 \to \text{No exact match} \\
&24 \to \text{No exact match} \\
&-103 \to \text{No exact match} \\
&-1 \to \text{No exact match} \\
&-8 \to \text{n. } -6 \quad (\text{Closest}) \\
&17.667 \to \text{No exact match} \\
&0 \to \text{No exact match} \\
\end{aligned}
\]
3. Decode the Message:
Using the matched letters:
\[
\text{h, c, d, n, l, j/k, b, n}
\]
This sequence does not form a clear word or phrase based on the given options. It seems there might be some rounding or approximation issues in matching the solutions to the answer key.
Final Answer:
\[
\boxed{\text{h, c, d, n, l, j/k, b, n}}
\]
If more context or clarification is needed, please provide additional details!
Parent Tip: Review the logic above to help your child master the concept of algebraic equations worksheets.
