To solve the problem, we will go through the following steps:
1. Rearrange the equation to find the initial cost of the bicycle:
The given equation is:
[tex]\[
y - 10 = -2(x - 10)
\][/tex]
We will rearrange this equation to the slope-intercept form [tex]\( y = mx + b \)[/tex].
Start by distributing the [tex]\(-2\)[/tex] on the right-hand side:
[tex]\[
y - 10 = -2x + 20
\][/tex]
Now, add 10 to both sides to isolate [tex]\(y\)[/tex]:
[tex]\[
y = -2x + 20 + 10
\][/tex]
Simplify the equation:
[tex]\[
y = -2x + 30
\][/tex]
2. Determine the initial cost of the bicycle:
When Hugo starts paying, the number of weeks [tex]\(x\)[/tex] is 0. Substitute [tex]\(x = 0\)[/tex] into the equation to find [tex]\(y\)[/tex]:
[tex]\[
y = -2(0) + 30
\][/tex]
[tex]\[
y = 30
\][/tex]
Therefore, the initial cost of the bicycle is \[tex]$30.
3. Calculate the number of weeks Hugo will take to finish paying for the bike:
To determine this, we need to find out after how many weeks \(x\) the amount of money owed, \(y\), becomes zero. Set \(y = 0\) and solve for \(x\):
\[
0 = -2x + 30
\]
Subtract 30 from both sides:
\[
-30 = -2x
\]
Divide both sides by -2:
\[
x = 15
\]
Hence, Hugo will finish paying for the bicycle after 15 weeks.
To summarize:
- The cost of the bicycle is \(\$[/tex]30\).
- Hugo will finish paying for the bike in 15 weeks.
