An airport has two long-term parking lots. The cost to park, [tex]\( y \)[/tex], in each lot for [tex]\( x \)[/tex] days is shown in the tables.

[tex]\[
\begin{tabular}{|c|c|}
\hline
\multicolumn{2}{|c|}{Lot A} \\
\hline
Days Parked & Cost \\
\hline
1 & \$16 \\
\hline
2 & \$20 \\
\hline
3 & \$24 \\
\hline
4 & \$28 \\
\hline
\end{tabular}
\][/tex]

[tex]\[
\begin{tabular}{|c|c|}
\hline
\multicolumn{2}{|c|}{Lot B} \\
\hline
Days Parked & Cost \\
\hline
1 & \$6 \\
\hline
2 & \$12 \\
\hline
3 & \$18 \\
\hline
4 & \$24 \\
\hline
\end{tabular}
\][/tex]

A system of linear equations can be used to determine on which day the cost to park is the same for both lots. One of the equations in the system is [tex]\( y = 6x \)[/tex]. What is the other equation in the system?



Answer :

To determine the equation for Lot A, we will start by observing the pattern in the given data.

The costs to park for different days for Lot A are as follows:
- On day 1, the cost is \[tex]$16. - On day 2, the cost is \$[/tex]20.
- On day 3, the cost is \[tex]$24. - On day 4, the cost is \$[/tex]28.

We can see that the cost increases by \[tex]$4 each additional day. To find the linear equation representing the cost \(y\) versus the number of days \(x\) for Lot A, we use the form of a linear equation: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. ### Step-by-Step Process 1. Determine the slope \(m\): - The slope \(m\) represents the rate of change in cost per day. - From the data, we notice the cost increases by \$[/tex]4 each day. Therefore, [tex]\(m = 4\)[/tex].

2. Determine the y-intercept [tex]\(b\)[/tex]:
- To find the y-intercept [tex]\(b\)[/tex], we use one of the points given in the data. Let's use the point (1, 16) (i.e., on day 1, the cost is \$16).
- Plugging this point into the equation [tex]\(y = mx + b\)[/tex]:
[tex]\[ 16 = 4(1) + b \\ 16 = 4 + b \\ b = 16 - 4 \\ b = 12 \][/tex]

3. Write the equation:
- Using the slope [tex]\(m = 4\)[/tex] and the y-intercept [tex]\(b = 12\)[/tex], we can write the equation for Lot A:
[tex]\[ y = 4x + 12 \][/tex]

Therefore, the other equation in the system, representing the cost to park in Lot A, is:
[tex]\[ y = 4x + 12 \][/tex]

For Lot B, we are already provided with the equation [tex]\(y = 6x\)[/tex].

Thus, the system of linear equations used to determine on which day the cost to park is the same for both lots is:
[tex]\[ \begin{cases} y = 6x \quad \text{(Lot B)} \\ y = 4x + 12 \quad \text{(Lot A)} \end{cases} \][/tex]