To solve the problem, we'll need to determine which of the provided equations will reach [tex]$0$[/tex] dollars in the same number of days as Mika, who has [tex]$\$[/tex]12[tex]$ and spends $[/tex]\[tex]$2$[/tex] per day.
Given the scenario, Mika's days to zero can be calculated as follows:
1. Mika's initial amount: \[tex]$12
2. Daily spend: \$[/tex]2
The number of days it takes for Mika to reach \[tex]$0 is:
\[ \text{Number of days} = \frac{12}{2} = 6 \text{ days} \]
Now we will calculate the number of days each given equation takes to reach \$[/tex]0.
### Equation: [tex]\( y = -2x - 12 \)[/tex]
1. At [tex]\( y = 0 \)[/tex]:
[tex]\[ 0 = -2x - 12 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ -2x = 12 \][/tex]
[tex]\[ x = -6 \][/tex]
Since -6 is not a valid number of days (days can't be negative), this equation does not match Mika's scenario.
### Equation: [tex]\( y = -3x + 18 \)[/tex]
1. At [tex]\( y = 0 \)[/tex]:
[tex]\[ 0 = -3x + 18 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ 3x = 18 \][/tex]
[tex]\[ x = 6 \][/tex]
So, it takes 6 days to reach [tex]\( y = 0 \)[/tex].
### Equation: [tex]\( y = -4x + 12 \)[/tex]
1. At [tex]\( y = 0 \)[/tex]:
[tex]\[ 0 = -4x + 12 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ 4x = 12 \][/tex]
[tex]\[ x = 3 \][/tex]
It takes 3 days to reach [tex]\( y = 0 \)[/tex].
### Equation: [tex]\( y = -12x + 2 \)[/tex]
1. At [tex]\( y = 0 \)[/tex]:
[tex]\[ 0 = -12x + 2 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ 12x = 2 \][/tex]
[tex]\[ x = \frac{2}{12} = \frac{1}{6} \][/tex]
It takes [tex]\(\frac{1}{6}\)[/tex] of a day to reach [tex]\( y = 0 \)[/tex].
### Conclusion:
The equation [tex]\( y = -3x + 18 \)[/tex] reaches \$0 in the same number of days (6 days) as Mika's scenario. Therefore, the relationship model that matches Mika's days is:
[tex]\[ \boxed{y = -3x + 18} \][/tex]
