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.
