Question

Kingwood Nail Salon is having a special this month on services. Over the weekend, they performed 31 manicures and 17 pedicures, bringing in a total of $955 in receipts. So far this week, they have administered 21 manicures and 39 pedicures, with receipts totalling $1,389. How much does the salon charge for each service?

Answer 1

Let "x" be the charge for manicures and let "y" be the charge for pedicures.

We can write the following system:

`\begin{gathered} 31x+17y=955 \\ 21x+39y=1389 \end{gathered}`

We're going to solve this system with the substitution method.

For this, we're going to solve the first equation for y:

`\begin{gathered} 17y=955-31x \\ y=(955-31x)/(17) \end{gathered}`

And now, we're going to replace this expression for y in the second equation to solve a linear equation which is easier:

`\begin{gathered} 21x+39y=1389 \\ 21x+39((955-31x)/(17))=1389 \end{gathered}`

Now, solving, we got:

`\begin{gathered} 21x+39((955-31x)/(17))=1389 \\ \\ 21x+(37245)/(17)-(1209)/(17)x=1389 \end{gathered}`

We could multiply the whole equation by 17 to make the process easier:

`\begin{gathered} 357x+37245-1209x=23613 \\ -852x=-13632 \\ x=16 \end{gathered}`

Therefore, x=16, so the charge for manicures is $16.

We're going to replace this value in any of both equations so we could find the value of y (The charge for pedicures)

`\begin{gathered} 31(16)+17y=955 \\ 17y=955-496 \\ 17y=459 \\ y=27 \end{gathered}`

Therefore, y=16, so the charge for pedicures is $27.