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Answer :
To determine how long it will take for Sylvia's loan balance to reach specific amounts, we can use the equation that models the loan balance over time. The equation given is:
[tex]\[ y = 382.05x + 25,077.40 \][/tex]
Here, [tex]\( y \)[/tex] represents the loan balance, and [tex]\( x \)[/tex] represents the number of months. We'll use this equation to find out how long it will take to reach certain loan balances by replacing [tex]\( y \)[/tex] with the target balances and solving for [tex]\( x \)[/tex].
### Let's consider two specific target balances and calculate the corresponding months:
1. Target Balance: \[tex]$15,829
We need to solve for \( x \) in the equation:
\[ 15,829 = 382.05x + 25,077.40 \]
Rearranging the equation to solve for \( x \):
\[ 382.05x = 15,829 - 25,077.40 \]
\[ 382.05x = -9,248.40 \]
\[ x = \frac{-9,248.40}{382.05} \]
After evaluating, we find that \( x \approx -24.21 \). This result suggests that the balance of \$[/tex]15,829 was reached 24.21 months before the start of the 24-month period provided.
2. Target Balance: \[tex]$16,633
Again, solving for \( x \) in the equation:
\[ 16,633 = 382.05x + 25,077.40 \]
Rearranging the equation to solve for \( x \):
\[ 382.05x = 16,633 - 25,077.40 \]
\[ 382.05x = -8,444.40 \]
\[ x = \frac{-8,444.40}{382.05} \]
After evaluating, we find that \( x \approx -22.10 \). This result suggests that the balance of \$[/tex]16,633 was reached 22.10 months before the start of the 24-month period provided.
Since both results are negative, it indicates these target balances were reached before our recorded time frame. If there are other balance targets you need help with, just let me know!
[tex]\[ y = 382.05x + 25,077.40 \][/tex]
Here, [tex]\( y \)[/tex] represents the loan balance, and [tex]\( x \)[/tex] represents the number of months. We'll use this equation to find out how long it will take to reach certain loan balances by replacing [tex]\( y \)[/tex] with the target balances and solving for [tex]\( x \)[/tex].
### Let's consider two specific target balances and calculate the corresponding months:
1. Target Balance: \[tex]$15,829
We need to solve for \( x \) in the equation:
\[ 15,829 = 382.05x + 25,077.40 \]
Rearranging the equation to solve for \( x \):
\[ 382.05x = 15,829 - 25,077.40 \]
\[ 382.05x = -9,248.40 \]
\[ x = \frac{-9,248.40}{382.05} \]
After evaluating, we find that \( x \approx -24.21 \). This result suggests that the balance of \$[/tex]15,829 was reached 24.21 months before the start of the 24-month period provided.
2. Target Balance: \[tex]$16,633
Again, solving for \( x \) in the equation:
\[ 16,633 = 382.05x + 25,077.40 \]
Rearranging the equation to solve for \( x \):
\[ 382.05x = 16,633 - 25,077.40 \]
\[ 382.05x = -8,444.40 \]
\[ x = \frac{-8,444.40}{382.05} \]
After evaluating, we find that \( x \approx -22.10 \). This result suggests that the balance of \$[/tex]16,633 was reached 22.10 months before the start of the 24-month period provided.
Since both results are negative, it indicates these target balances were reached before our recorded time frame. If there are other balance targets you need help with, just let me know!
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