Certainly, let's solve the problem step-by-step.
1. How much did the bicycle cost?
The given equation for the amount of money Hugo owes is:
[tex]\[y - 10 = -2(x - 10)\][/tex]
To find the initial cost of the bicycle, we need to determine the amount Hugo owed initially (week 0). This means we set [tex]\( x = 0 \)[/tex] because that represents the start before any payments were made.
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]\[ y = 20 + 10 \][/tex]
[tex]\[ y = 30 \][/tex]
So, the initial cost of the bicycle was \[tex]$30.
2. After how many weeks will Hugo finish paying for the bike?
Hugo will have finished paying for the bike when the amount of money he owes (\( y \)) is equal to 0.
Set \( y = 0 \) in the given equation and solve for \( x \):
\[ 0 - 10 = -2(x - 10) \]
\[ -10 = -2(x - 10) \]
Divide both sides by -2:
\[ 5 = x - 10 \]
\[ x = 5 + 10 \]
\[ x = 15 \]
So, Hugo will finish paying for the bike after 15 weeks.
3. Create the Graph
To graph the equation \( y - 10 = -2(x - 10) \), we can determine a few points to plot.
\[
\begin{array}{|c|c|}
\hline
x & y \\ \hline
5 & 20 \\ \hline
10 & 10 \\ \hline
15 & 0 \\ \hline
\end{array}
\]
- At \( x = 5 \):
\[ y - 10 = -2(5 - 10) \]
\[ y - 10 = 10 \]
\[ y = 20 \]
- At \( x = 10 \):
\[ y - 10 = -2(10 - 10) \]
\[ y - 10 = 0 \]
\[ y = 10 \]
- At \( x = 15 \):
\[ y - 10 = -2(15 - 10) \]
\[ y - 10 = -10 \]
\[ y = 0 \]
These points (\(x, y\)) are (5, 20), (10, 10), and (15, 0).
You can plot these points on a graph, and draw a line through them to visualize how the amount Hugo owes decreases over time.
Conclusion:
- The cost of the bicycle is \$[/tex]30.
- Hugo will finish paying for the bike in 15 weeks.
Here’s the table for the graph:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\ \hline
5 & 20 \\ \hline
10 & 10 \\ \hline
15 & 0 \\ \hline
\end{array}
\][/tex]
