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Sagot :
Let's analyze each choice to determine the correct system of equations that accurately represents Regina's financial situation.
1. Initial Setup:
- Regina's initial savings: \[tex]$500 - Cost of the washer and dryer: \$[/tex]1,600
- Monthly loan payment: \[tex]$134 - Additional savings each month: \$[/tex]75
2. Understanding the Equations:
- The first equation should represent Regina's growing savings over time, taking into account her additional monthly savings.
- The second equation should represent the remaining balance on the washer and dryer loan after each monthly payment.
### Evaluating Each Option
Option A:
[tex]\[ \begin{aligned} y & = 134x + 500 \\ y & = 1,600 - 100x \end{aligned} \][/tex]
- First equation [tex]\(y = 134x + 500\)[/tex]: This means Regina saves \[tex]$134 each month plus an initial \$[/tex]500 saving. However, this doesn't include her additional savings of \[tex]$75 each month - Incorrect. - Second equation \(y = 1,600 - 100x\): This attempts to represent her remaining loan balance, but the monthly deduction incorrectly uses 100 instead of 134 - Incorrect. Option B: \[ \begin{aligned} y & = 75x + 500 \\ y & = 1,600 - 134x \end{aligned} \] - First equation \(y = 75x + 500\): This correctly includes her initial savings of \$[/tex]500 and her extra savings of \[tex]$75 each month - Correct. - Second equation \(y = 1,600 - 134x\): This correctly represents the remaining cost of the washer and dryer after deducting the monthly payment of \$[/tex]134 - Correct.
Option C:
[tex]\[ \begin{aligned} y & = 500x + 75 \\ y & = 1,600 - 134x \end{aligned} \][/tex]
- First equation [tex]\(y = 500x + 75\)[/tex]: This incorrectly implies that she saves \[tex]$500 each month plus a one-time \$[/tex]75. This is not consistent with her actual additional savings each month - Incorrect.
- Second equation [tex]\(y = 1,600 - 134x\)[/tex]: This correctly represents the remaining loan balance - Correct.
Option D:
[tex]\[ \begin{aligned} y & = 1,600x + 134 \\ y & = 75x - 500 \end{aligned} \][/tex]
- First equation [tex]\(y = 1,600x + 134\)[/tex]: This doesn't align with any logical financial scenario in this context as it implies a massive monthly saving which is unrealistic - Incorrect.
- Second equation [tex]\(y = 75x - 500\)[/tex]: This incorrectly represents a negative initial balance and savings while the monthly saving is correctly included - Incorrect.
### Conclusion:
After evaluating each option and comparing with Regina's financial details, Option B correctly represents Regina’s savings growth and the remaining loan balance:
[tex]\[ \begin{aligned} y & = 75x + 500 \\ y & = 1,600 - 134x \end{aligned} \][/tex]
Thus, the correct answer is B.
1. Initial Setup:
- Regina's initial savings: \[tex]$500 - Cost of the washer and dryer: \$[/tex]1,600
- Monthly loan payment: \[tex]$134 - Additional savings each month: \$[/tex]75
2. Understanding the Equations:
- The first equation should represent Regina's growing savings over time, taking into account her additional monthly savings.
- The second equation should represent the remaining balance on the washer and dryer loan after each monthly payment.
### Evaluating Each Option
Option A:
[tex]\[ \begin{aligned} y & = 134x + 500 \\ y & = 1,600 - 100x \end{aligned} \][/tex]
- First equation [tex]\(y = 134x + 500\)[/tex]: This means Regina saves \[tex]$134 each month plus an initial \$[/tex]500 saving. However, this doesn't include her additional savings of \[tex]$75 each month - Incorrect. - Second equation \(y = 1,600 - 100x\): This attempts to represent her remaining loan balance, but the monthly deduction incorrectly uses 100 instead of 134 - Incorrect. Option B: \[ \begin{aligned} y & = 75x + 500 \\ y & = 1,600 - 134x \end{aligned} \] - First equation \(y = 75x + 500\): This correctly includes her initial savings of \$[/tex]500 and her extra savings of \[tex]$75 each month - Correct. - Second equation \(y = 1,600 - 134x\): This correctly represents the remaining cost of the washer and dryer after deducting the monthly payment of \$[/tex]134 - Correct.
Option C:
[tex]\[ \begin{aligned} y & = 500x + 75 \\ y & = 1,600 - 134x \end{aligned} \][/tex]
- First equation [tex]\(y = 500x + 75\)[/tex]: This incorrectly implies that she saves \[tex]$500 each month plus a one-time \$[/tex]75. This is not consistent with her actual additional savings each month - Incorrect.
- Second equation [tex]\(y = 1,600 - 134x\)[/tex]: This correctly represents the remaining loan balance - Correct.
Option D:
[tex]\[ \begin{aligned} y & = 1,600x + 134 \\ y & = 75x - 500 \end{aligned} \][/tex]
- First equation [tex]\(y = 1,600x + 134\)[/tex]: This doesn't align with any logical financial scenario in this context as it implies a massive monthly saving which is unrealistic - Incorrect.
- Second equation [tex]\(y = 75x - 500\)[/tex]: This incorrectly represents a negative initial balance and savings while the monthly saving is correctly included - Incorrect.
### Conclusion:
After evaluating each option and comparing with Regina's financial details, Option B correctly represents Regina’s savings growth and the remaining loan balance:
[tex]\[ \begin{aligned} y & = 75x + 500 \\ y & = 1,600 - 134x \end{aligned} \][/tex]
Thus, the correct answer is B.
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