Miss Mendoza is getting married soon. She would like to hire a professional who will come to her house and do hair and makeup for the bridal party on her wedding day. She has narrowed it down to two possible choices. A woman from Winchester Salon charges an initial fee of $60 and then an additional $10 per hour. Alternatively, a team from Beauty by Nellie offers a free consultation and then charges a higher hourly rate of $30. If it takes a certain amount of time to finish everyone's hair and makeup, Miss Mendoza will end up paying the same amount either way. How much would Miss Mendoza end up paying?
Write a system of equations, graph them, and type the solution.

To solve the problem, we need to find the amount of time $ t $ in hours that results in the same total cost for both options. We will set up a system of equations to represent the total cost for each choice and then solve for $ t $.

1. **Cost for Winchester Salon:**
   - Initial fee: \$60
   - Hourly rate: \$10 per hour

   The total cost for $ t $ hours is:
   \[
   C_1 = 60 + 10t
   \]

2. **Cost for Beauty by Nellie:**
   - Initial fee: \$0 (free consultation)
   - Hourly rate: \$30 per hour

   The total cost for $ t $ hours is:
   \[
   C_2 = 30t
   \]

We need to find the time $ t $ when the costs are equal:
\[
60 + 10t = 30t
\]

**Solving the equation:**

Subtract $ 10t $ from both sides:
\[
60 = 20t
\]

Divide both sides by 20:
\[
t = 3
\]

So, the time $ t $ at which the costs are equal is 3 hours. To find the total cost Miss Mendoza would end up paying, we can substitute $ t = 3 $ back into either cost equation:

Using $ C_1 = 60 + 10t $:
\[
C_1 = 60 + 10(3) = 60 + 30 = 90
\]

Using $ C_2 = 30t $:
\[
C_2 = 30(3) = 90
\]

Therefore, Miss Mendoza would end up paying \$90.

**Graphical Representation:**

To graph the equations $ C_1 = 60 + 10t $ and $ C_2 = 30t $:

1. The y-intercept of $ C_1 $ is 60 and the slope is 10.
2. The y-intercept of $ C_2 $ is 0 and the slope is 30.

The lines intersect at $ t = 3 $, and $ C = 90 $.

Here is the graph of the system of equations:

\[
\begin{array}{l}
\text{Graph of } C_1 = 60 + 10t \\
\text{Graph of } C_2 = 30t \\
\end{array}
\]

![Graph of the System](https://via.placeholder.com/300x300.png?text=Graph+of+Equations)

The intersection point $(3, 90)$ confirms that both cost functions are equal at $ t = 3 $ hours and the cost is \$90.

Do you want details or have any questions?

**Relative Questions:**

1. What if Winchester Salon changes its initial fee to \$70?
2. How would the solution change if Beauty by Nellie increased their hourly rate to \$35?
3. What would be the new cost if the time needed increased to 4 hours for both services?
4. How does the solution change if Beauty by Nellie starts charging an initial fee of \$20?
5. What if Miss Mendoza gets a discount of \$10 from Winchester Salon on the total cost?
6. How would the graph look if the hourly rate for Winchester Salon decreases to \$8?
7. If the consultation at Beauty by Nellie becomes \$10, how does it affect the cost?
8. What if Winchester Salon offers a package deal of \$100 for up to 5 hours?

**Tip:**

When comparing services with different pricing structures, setting up equations and solving for the point where they intersect helps determine the best choice for given conditions.

Explore how to calculate costs for hair and makeup services based on different pricing models. Solve equations and graph cost functions to determine optimal choices. Suitable for high school students learning about linear equations and graphing.

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