Question

A locksmith charges $25 to make a house call and $15 for each lock that is re-keyed. Another locksmith charges $10 to make a house call and $20 for each lock that is re keyed Determine the correct system of equations

Answer 1

We will investigate the methodolgy for developing a system of linear equations based on data given.

We will first define the variables pertaining to number of locks re-keyed by each of the locksmith as follows:

`\begin{gathered} \text{Locksmith ( A ) = x locks re-keyed} \\ \text{Locksmith ( B ) = y locks re-keyed} \end{gathered}`

Now we will express the rates charged by each locksmith for a house-call. This is expressed as a fixed base rate set by each locksmith on their own.

`\begin{gathered} \text{House-call charge ( A ) = \$25 } \\ \text{House-call charge ( B ) = \$}10 \end{gathered}`

Next we will express the rates charged per each number of locks re-keyed by each of the locksmith as follows:

`\begin{gathered} \text{Locks rate ( A ) = \$15 for each lock} \\ \text{Locks rate ( B ) = \$20 for each lock} \end{gathered}`

The above charges are variable charges i.e they depend on the total number of locks re-keyed by each locksmith. The Job charges can then be calculated as:

`\begin{gathered} \text{Job charge = Locks rate }\cdot\text{ Locks rekeyed} \\ \\ \text{Job charge ( A ) = 15 }\cdot\text{ x} \\ \text{Job charge ( B ) = 20 }\cdot\text{ y} \end{gathered}`

The total receipt charged by each locksmith for any house-call job can be expressed in terms of the House-call rate and the amount charged for the job as follows:

`\begin{gathered} \text{Total Receipt = House-call charge + Job charge} \\ \\ \text{\textcolor{#FF7968}{Total Receipt ( A ) = 25 + 15x}} \\ \text{\textcolor{#FF7968}{Total Receipt ( B ) = 10 + 20y}} \end{gathered}`

The above total receipts charged by each of the locksmith serves as a system of equations:

`\begin{gathered} \text{Total Receipt ( A ) = 25 + 15x} \\ \text{Total Receipt ( B ) = 10 + 20y} \end{gathered}`