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Hugo decided to pay [tex]$2$[/tex] a week to his brother to buy a bicycle. The equation [tex]$y - 10 = -2(x - 10)$[/tex] models the amount of money he owes for the bike, where [tex]$x$[/tex] is the number of weeks and [tex]$y$[/tex] is the amount of money he still needs to pay.

1. How much did the bicycle cost?
[tex]$\$[/tex][tex]$ $[/tex]\square[tex]$

2. After how many weeks will Hugo finish paying for the bike?
$[/tex]\square[tex]$ weeks

3. Graph: $[/tex]y - 10 = -2(x - 10)[tex]$
\begin{tabular}{|l|l|}
\hline
$[/tex]x[tex]$ & $[/tex]y$ \\
\hline
& \\
& \\
& \\
\hline
\end{tabular}

Click or tap the graph to plot a point.

Sagot :

GinnyAnswer
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]
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