Question

miss dalton 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 cedarburg salon charges an initial fee of $30 and then an additional $25 per hour. alternatively, a team from beauty by nolan offers a free consultation and then charges a higher hourly rate of $40. if it takes a certain amount of time to finish everyones hair and makeup, miss dalton will end up paying the same amount either way. how much would miss dalton end up paying? write a system of equations, graph them, and type the solution.

Explanation:

Step1: Define variables and equations

Let \(x\) be the number of hours and \(y\) be the total amount paid.
For Cedarburg Salon: \(y = 25x+30\) (initial fee of \(30\) and \(25\) per hour).
For Beauty by Nolan: \(y = 40x\) (free consultation, \(40\) per hour).

Step2: Set the equations equal to each other

Set \(25x + 30=40x\).
Subtract \(25x\) from both sides: \(30=40x - 25x\).
Simplify: \(30 = 15x\).
Solve for \(x\): \(x=\frac{30}{15}=2\).

Step3: Find the value of \(y\)

Substitute \(x = 2\) into \(y = 40x\) (or \(y=25x + 30\)).
Using \(y = 40x\), we get \(y=40\times2=80\).

Answer:

The system of equations is \(

$$\begin{cases}y = 25x+30\\y = 40x\end{cases}$$

\). The solution (point of intersection) is \((2,80)\). So Miss Dalton would end up paying \(80\) dollars.