Let's analyze the given equation and questions step by step.
### Step 1: Determine the initial cost of the bicycle
The given equation is:
[tex]\[ y - 10 = -2(x - 10) \][/tex]
First, we need to find the amount of money Hugo owes initially, which is at [tex]\(x = 0\)[/tex] weeks.
Substitute [tex]\(x = 0\)[/tex] into the equation:
[tex]\[ y - 10 = -2(0 - 10) \][/tex]
[tex]\[ y - 10 = -2(-10) \][/tex]
[tex]\[ y - 10 = 20 \][/tex]
[tex]\[ y = 30 \][/tex]
So, the initial cost of the bicycle is:
[tex]\[ \text{\$} 30 \][/tex]
### Step 2: Find out after how many weeks Hugo will finish paying for the bike
To determine when Hugo will finish paying for the bike, we need to find when the remaining amount [tex]\( y = 0 \)[/tex].
Set [tex]\( y = 0 \)[/tex] in the equation:
[tex]\[ 0 - 10 = -2(x - 10) \][/tex]
[tex]\[ -10 = -2(x - 10) \][/tex]
Divide both sides by [tex]\(-2\)[/tex]:
[tex]\[ 5 = x - 10 \][/tex]
[tex]\[ x = 15 \][/tex]
So, Hugo will finish paying for the bike after:
[tex]\[ 15 \text{ weeks} \][/tex]
### Step 3: Fill in the table for the graph
We need to fill the table with values of [tex]\( x \)[/tex] and corresponding [tex]\( y \)[/tex] values. Let's use the points [tex]\( x = 0, 5, 10, 15, 20 \)[/tex].
1. For [tex]\( x = 0 \)[/tex]:
[tex]\[ y - 10 = -2(0 - 10) \rightarrow y - 10 = 20 \rightarrow y = 30 \][/tex]
2. For [tex]\( x = 5 \)[/tex]:
[tex]\[ y - 10 = -2(5 - 10) \rightarrow y - 10 = 10 \rightarrow y = 20 \][/tex]
3. For [tex]\( x = 10 \)[/tex]:
[tex]\[ y - 10 = -2(10 - 10) \rightarrow y - 10 = 0 \rightarrow y = 10 \][/tex]
4. For [tex]\( x = 15 \)[/tex]:
[tex]\[ y - 10 = -2(15 - 10) \rightarrow y - 10 = -10 \rightarrow y = 0 \][/tex]
5. For [tex]\( x = 20 \)[/tex]:
[tex]\[ y - 10 = -2(20 - 10) \rightarrow y - 10 = -20 \rightarrow y = -10 \][/tex]
### Filled table:
| [tex]\( x \)[/tex] | [tex]\( y \)[/tex] |
| :--: | :--: |
| 0 | 30 |
| 5 | 20 |
| 10 | 10 |
| 15 | 0 |
| 20 | -10 |
### Summary
- Initial cost of the bicycle: \$30
- Weeks required to finish paying: 15 weeks
- Graph table:
[tex]\[
\begin{tabular}{|c|c|}
\hline
\( x \) & \( y \) \\
\hline
0 & 30 \\
\hline
5 & 20 \\
\hline
10 & 10 \\
\hline
15 & 0 \\
\hline
20 & -10 \\
\hline
\end{tabular}
\][/tex]
