Let's solve this step-by-step.
### 1. Determining the Initial Cost of the Bicycle
We start with the equation given:
[tex]\[ y - 10 = -2(x - 10) \][/tex]
To find the initial cost of the bicycle, we need to determine the amount Hugo owed initially, which corresponds to when [tex]\( x = 0 \)[/tex] (the beginning). Substituting [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]\(\$30\)[/tex].
Answer: The bicycle cost [tex]\(\$30\)[/tex].
### 2. Determining After How Many Weeks Hugo Will Finish Paying for the Bike
To find out when Hugo finishes paying, we need to know when the amount he owes [tex]\( y \)[/tex] becomes 0. Thus, we set [tex]\( y = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 - 10 = -2(x - 10) \][/tex]
[tex]\[ -10 = -2(x - 10) \][/tex]
[tex]\[ -10 = -2x + 20 \][/tex]
[tex]\[ -10 - 20 = -2x \][/tex]
[tex]\[ -30 = -2x \][/tex]
[tex]\[ x = 15 \][/tex]
So, Hugo will finish paying for the bike after 15 weeks.
Answer: Hugo will finish paying for the bike after 15 weeks.
### 3. Completing the Graph
We need to plot the points that satisfy the equation [tex]\( y - 10 = -2(x - 10) \)[/tex] on a graph.
Let's create a few points using the equation:
#### Example Points:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ y - 10 = -2(0 - 10) \][/tex]
[tex]\[ y - 10 = 20 \][/tex]
[tex]\[ y = 30 \][/tex]
[tex]\((0, 30)\)[/tex]
- For [tex]\( x = 10 \)[/tex]:
[tex]\[ y - 10 = -2(10 - 10) \][/tex]
[tex]\[ y - 10 = 0 \][/tex]
[tex]\[ y = 10 \][/tex]
[tex]\((10, 10)\)[/tex]
- For [tex]\( x = 15 \)[/tex]:
[tex]\[ y - 10 = -2(15 - 10) \][/tex]
[tex]\[ y - 10 = -10 \][/tex]
[tex]\[ y = 0 \][/tex]
[tex]\((15, 0)\)[/tex]
Let's fill in the graph with these points:
### Table of Values:
\begin{tabular}{|l|l|}
\hline[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline 0 & 30 \\
\hline 10 & 10 \\
\hline 15 & 0 \\
\hline
\end{tabular}
With these points, you can plot the line [tex]\( y - 10 = -2(x - 10) \)[/tex]. So, you would start at [tex]\((0, 30)\)[/tex], cross [tex]\((10, 10)\)[/tex], and end at [tex]\((15, 0)\)[/tex].
