Question: Mia charges a $20 starting fee plus $12 per hour for babysitting. Emma charges no starting fee but charges $15 per hour.
Write a system of equations to represent this relationship. Graph the equations and use the graph to determine the number of hours that would cost the same for both babysitters.
Be sure to use let statements to define variables!
1. **State the problem:**
We want to find the number of hours $h$ where the total cost for Mia and Emma is the same.
2. **Define variables:**
Let $h$ = number of hours babysitting.
Let $C_M$ = total cost for Mia.
Let $C_E$ = total cost for Emma.
3. **Write the equations:**
Mia charges a $20 starting fee plus $12 per hour:
$$C_M = 20 + 12h$$
Emma charges no starting fee but $15 per hour:
$$C_E = 15h$$
4. **Set the costs equal to find when they charge the same:**
$$20 + 12h = 15h$$
5. **Solve for $h$:**
Subtract $12h$ from both sides:
$$20 + \cancel{12h} = 15h - \cancel{12h}$$
$$20 = 3h$$
Divide both sides by 3:
$$\frac{20}{\cancel{3}} = \frac{3h}{\cancel{3}}$$
$$h = \frac{20}{3} \approx 6.67$$
6. **Interpretation:**
At approximately $6.67$ hours, the cost for both babysitters is the same.
7. **Summary:**
- Mia's cost equation: $$C_M = 20 + 12h$$
- Emma's cost equation: $$C_E = 15h$$
- Equal cost at $$h = \frac{20}{3}$$ hours.
This means if you babysit for about 6 hours and 40 minutes, both Mia and Emma will charge the same total amount.