Let's analyze the different systems of equations to determine which can be used to find when Rafiki's savings will equal the remainder of the loan.
### Understanding the Variables
- [tex]\( x \)[/tex] represents the number of months.
- [tex]\( y \)[/tex] represents the dollar value of each account.
### Systems of Equations
1. System 1:
[tex]\[
\begin{array}{l}
y = 100x + 200 \\
y = 2,000 - 25x
\end{array}
\][/tex]
2. System 2:
[tex]\[
\begin{array}{l}
y = 200x + 100 \\
y = 2,000 - 200x
\end{array}
\][/tex]
3. System 3:
[tex]\[
y = 100x + 200 \\
y = 2,000 - 75x
\][/tex]
4. System 4:
[tex]\[
\begin{array}{l}
y = 100x + 200 \\
y = 2,300 - 75x
\end{array}
\][/tex]
### Analyzing Each System
System 1 Analysis:
- Equation 1: [tex]\( y = 100x + 200 \)[/tex]
This can represent savings accumulating from a base of [tex]$200 at a rate of $[/tex]100 per month.
- Equation 2: [tex]\( y = 2,000 - 25x \)[/tex]
This looks like a loan balance starting at [tex]$2,000 and decreasing by $[/tex]25 each month. However, Rafiki's loan is paid off at [tex]$75 each month, not $[/tex]25.
System 2 Analysis:
- Equation 1: [tex]\( y = 200x + 100 \)[/tex]
This suggests savings accumulating, but the rate and base don't seem to match likely scenarios.
- Equation 2: [tex]\( y = 2,000 - 200x \)[/tex]
This decreases too quickly ([tex]$200 per month) compared to the given loan repayable at $[/tex]75 per month.
System 3 Analysis:
- Equation 1: [tex]\( y = 100x + 200 \)[/tex]
This suggests savings start from [tex]$200 and increase by $[/tex]100 every month.
- Equation 2: [tex]\( y = 2,000 - 75x \)[/tex]
This equation indicates the loan starts at [tex]$2,000 and decreases by $[/tex]75 each month, matching the given payment structure.
System 4 Analysis:
- Equation 1: [tex]\( y = 100x + 200 \)[/tex]
This setup matches the savings structure.
- Equation 2: [tex]\( y = 2,300 - 75x \)[/tex]
This indicates the loan balance starts at [tex]$2,300 and decreases by $[/tex]75 each month, which doesn't match $2,000 as the initial loan amount.
### Conclusion
The correct system of equations that can be used to solve for the time it will take for Rafiki's savings to equal the remainder of the loan is:
[tex]\[
y = 100x + 200 \\
y = 2,000 - 75x
\][/tex]
