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Answer :
Sure! Let's determine how long it will take for Sylvia's loan balance to reach around [tex]$17,000 and $[/tex]0 using the best-fit equation provided. The equation modeling the data is:
[tex]\[ y = -382.05x + 25,077.40 \][/tex]
Step-by-Step Solution:
1. Find the time to reach a balance of [tex]$17,000:
- Set the equation for \( y \) equal to $[/tex]17,000:
[tex]\[
17,000 = -382.05x + 25,077.40
\][/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[
382.05x = 25,077.40 - 17,000
\][/tex]
[tex]\[
382.05x = 8,077.40
\][/tex]
[tex]\[
x = \frac{8,077.40}{382.05} \approx 22.97
\][/tex]
This calculation shows that it will take approximately 23 months for the loan balance to be around [tex]$17,000.
2. Find the time to reach a balance of $[/tex]0:
- Set the equation for [tex]\( y \)[/tex] equal to [tex]$0:
\[
0 = -382.05x + 25,077.40
\]
- Solve for \( x \):
\[
382.05x = 25,077.40
\]
\[
x = \frac{25,077.40}{382.05} \approx 67.47
\]
This calculation shows that it will take approximately 67 months for the loan balance to reach $[/tex]0.
Answer:
- The balance of the loan will be approximately [tex]$17,000 at around month 23.
- The balance of the loan will reach $[/tex]0 at around month 67.
[tex]\[ y = -382.05x + 25,077.40 \][/tex]
Step-by-Step Solution:
1. Find the time to reach a balance of [tex]$17,000:
- Set the equation for \( y \) equal to $[/tex]17,000:
[tex]\[
17,000 = -382.05x + 25,077.40
\][/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[
382.05x = 25,077.40 - 17,000
\][/tex]
[tex]\[
382.05x = 8,077.40
\][/tex]
[tex]\[
x = \frac{8,077.40}{382.05} \approx 22.97
\][/tex]
This calculation shows that it will take approximately 23 months for the loan balance to be around [tex]$17,000.
2. Find the time to reach a balance of $[/tex]0:
- Set the equation for [tex]\( y \)[/tex] equal to [tex]$0:
\[
0 = -382.05x + 25,077.40
\]
- Solve for \( x \):
\[
382.05x = 25,077.40
\]
\[
x = \frac{25,077.40}{382.05} \approx 67.47
\]
This calculation shows that it will take approximately 67 months for the loan balance to reach $[/tex]0.
Answer:
- The balance of the loan will be approximately [tex]$17,000 at around month 23.
- The balance of the loan will reach $[/tex]0 at around month 67.
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