Question

Write a system of equations to describe the situation below, solve using an augmented matrix.Oakland Nail Salon is having a special this month on services. Over the weekend, they performed 14 manicures and 1 pedicure, bringing in a total of $206 in receipts. So far this week, they have administered 31 manicures and 1 pedicure, with receipts totalling $427. How much does the salon charge for each service?The salon charges $? for a manicure and $? for a pedicure.

Answer 1

Given that they performed 14 manicures and 1 pedicure, bringing in a total of $206 in receipts. and they have administered 31 manicures and 1 pedicure, with receipts totalling $427. The salon charges for a manicure and for a pedicure can be derived below.

Explanation

Let the charges for a manicure and for a pedicure be x and y respectively

Therefore, the system of equations can be derived as;

`\begin{gathered} 14x+y=206----1 \\ 31x+y=427----2 \end{gathered}`

In augumented matrix form, this can be expressed as;

`\begin{bmatrix}{14} & {1} & {206} \\ {31} & {1} & {427} \\ {} & {} & {}\end{bmatrix}`

Here we perform the row operation:

R1→ R1 - R2

`\begin{bmatrix}{-17} & {0} & {-221} \\ {31} & {1} & {427} \\ {} & {} & {}\end{bmatrix}`

R1→ 1/-17.R1

`\begin{bmatrix}{1} & {0} & {13} \\ {31} & {1} & {427} \\ {} & {} & {}\end{bmatrix}`

R2→ R2 - 31R1

`\begin{bmatrix}{1} & {0} & {13} \\ {0} & {1} & 24 \\ {} & {} & {}\end{bmatrix}`

Thus the last column represents the values of the variables and we have x = 13, and y = 24

Answer: Therefore, the salon charges 13 dollars for a manicure and 24 dollars for a pedicure