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Graham needs to get a ride home from the airport. He wants to use a rideshare company but needs to choose between two different companies.

Company [tex]$A$[/tex] charges a flat fee of [tex]$\$[/tex] 8[tex]$ plus $[/tex]\[tex]$ 0.25$[/tex] per mile.
Company [tex]$B$[/tex] charges a flat fee of [tex]$\$[/tex] 6[tex]$ plus $[/tex]\[tex]$ 0.60$[/tex] per mile.

Let [tex]$x$[/tex] represent the number of miles of a ride, and let [tex]$y$[/tex] represent the total price paid for a ride. The system of equations below represents this situation:

Company [tex]$A$[/tex]: [tex] y = 0.25x + 8 [/tex]
Company [tex]$B$[/tex]: [tex] y = 0.60x + 6 [/tex]

What does the solution to the system of equations represent in the context of the situation the system models?

A. The solution represents the number of miles that will result in a ride with both companies costing the same total amount.
B. The solution represents the rideshare companies charging the same amount for each mile of a ride.
C. The solution represents how many miles it is from the airport to Graham's house.
D. The solution represents the rideshare companies charging the flat fee for a ride.



Answer :

GinnyAnswer
Let's solve the system of equations to determine the point at which both companies charge the same total amount. The equations given for the two companies are:

For Company A:
[tex]\[ y = 0.25z + 8 \][/tex]

For Company B:
[tex]\[ y = 0.60z + 6 \][/tex]

To find the number of miles \( z \) at which both companies charge the same price, we set the two equations equal to each other:
[tex]\[ 0.25z + 8 = 0.60z + 6 \][/tex]

Next, we isolate the variable \( z \). First, subtract \( 0.25z \) from both sides:
[tex]\[ 8 = 0.35z + 6 \][/tex]

Then, subtract 6 from both sides:
[tex]\[ 2 = 0.35z \][/tex]

Now, divide both sides by \( 0.35 \):
[tex]\[ z = \frac{2}{0.35} \][/tex]

After performing the division, we get:
[tex]\[ z \approx 5.714 \][/tex]

Therefore, the number of miles that will result in a ride costing the same amount for both companies is approximately \( 5.714 \) miles.

In the context of the situation, the solution represents the number of miles at which both rideshare companies will charge Graham the same total amount for the ride from the airport to his destination.

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