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Chris purchased a new mattress for [tex]$\$[/tex]1,399[tex]$. He will need to pay $[/tex]\[tex]$78$[/tex] each month on his interest-free loan to pay off the total price of the mattress in 18 months. Separately, Chris sets aside [tex]$\$[/tex]50[tex]$ each month in a savings account, which currently shows a balance of $[/tex]\[tex]$250$[/tex].

Create an augmented matrix from a system of equations to help Chris determine when he can use monthly payments and savings to pay off the total purchase price of the mattress.

[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
& Column 1 & Column 2 & Column 3 \\
\hline
Row 1 & Q & 0 & 0 \\
\hline
Row 2 & 0 & & \\
\hline
& & & \\
\hline
\end{tabular}
\][/tex]



Answer :

GinnyAnswer
Sure, let's break this down step-by-step.

1. Monthly Payment Calculation:
- Chris needs to pay [tex]$\$[/tex] 78[tex]$ per month. - Over 18 months, the total payment would be \( 78 \times 18 \). 2. Total Savings Calculation: - Chris saves $[/tex]\[tex]$ 50$[/tex] per month.
- Over 18 months, his total savings from the monthly savings would be [tex]\( 50 \times 18 \)[/tex].
- Adding the initial savings of [tex]$\$[/tex] 250$, the total savings after 18 months would be [tex]\( 50 \times 18 + 250 \)[/tex].

3. Creating the Augmented Matrix:
- We represent the system of equations in matrix form.
- The equation for the monthly payments:
[tex]\( 78x + 0y = 1399 \)[/tex]
- The equation for the savings:
[tex]\( 0x + 1y = \text{Total Savings After 18 months} \)[/tex]
- Here, [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are placeholders representing the multiplication factors for monthly payments and savings.

Given these points, let's construct the augmented matrix:
- The first row represents the monthly payments:
[tex]\[ 78 \quad 0 \quad 1399 \][/tex]
- The second row represents the total savings:
[tex]\[ 0 \quad 1 \quad 1150 \][/tex]

Therefore, the completed augmented matrix becomes:

\begin{tabular}{|c|c|c|c|}
\hline
& Column 1 & Column 2 & Column 3 \\
\hline
Row 1 & 78 & 0 & 1399 \\
\hline
Row 2 & 0 & 1 & 1150 \\
\hline
\end{tabular}

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