To determine which equations correctly model the total fare [tex]\( F \)[/tex] for [tex]\( m \)[/tex] miles driven by a taxi service that charges a flat rate of [tex]$15 plus $[/tex]2 per mile, let's carefully analyze the given options:
1. Option 1: [tex]\( F = 15m + 2 \)[/tex]
- This equation suggests that the total fare increases by [tex]$15 per mile and adds a flat rate of $[/tex]2.
- This is incorrect because the fare should increase by [tex]$2 per mile and have a flat rate of $[/tex]15.
2. Option 2: [tex]\( F = 15 + 2m \)[/tex]
- Here, the equation correctly adds a flat rate of [tex]$15 and then increases the fare by $[/tex]2 for each mile driven.
- This correctly represents the situation where the total fare [tex]\( F \)[/tex] is composed of a fixed cost of [tex]$15 plus an additional $[/tex]2 for each mile driven.
3. Option 3: [tex]\( F = 2 + 15m \)[/tex]
- This equation suggests that the total fare increases by [tex]$15 per mile and adds a flat rate of $[/tex]2.
- This is incorrect because, as with Option 1, the fare should increase by [tex]$2 per mile and have a flat rate of $[/tex]15.
4. Option 4: [tex]\( F = 2m + 15 \)[/tex]
- This equation adds the per-mile cost of [tex]$2 for each mile (\( 2m \)) and then a flat rate of $[/tex]15.
- This correctly represents the scenario where the fare is a sum of a [tex]$15 flat rate and an additional $[/tex]2 per mile driven.
Therefore, the correct equations that model the total fare [tex]\( F \)[/tex] for [tex]\( m \)[/tex] miles driven are:
[tex]\[ F = 15 + 2m \][/tex]
[tex]\[ F = 2m + 15 \][/tex]
Thus, the correct options are:
- [tex]\( \boxed{2} \)[/tex]
- [tex]\( \boxed{4} \)[/tex]
