Example

A business wants to manufacture bike frames. Before they start production, they need to make sure they can make a profit with the materials and labor force they have. Their accountant has given them a cost equation of [latex]y=0.85x+35,000[/latex] and a revenue equation of [latex]y=1.55x[/latex]:

1. Interpret [latex]x[/latex] and [latex]y[/latex] for the cost equation
2. Interpret [latex]x[/latex] and [latex]y[/latex] for the revenue equation

Answer: Cost: [latex]y=0.85x+35,000[/latex] Revenue:[latex]y=1.55x[/latex] The cost equation represents money leaving the company, namely how much it costs to produce a given number of bike frames. If we use the skateboard example as a model, [latex]x[/latex] would represent the number of frames produced (instead of skateboards) and y would represent the amount of money it would cost to produce them (the same as the skateboard problem). The revenue equation represents money coming into the company, so in this context [latex]x[/latex] still represents the number of bike frames manufactured, and [latex]y[/latex] now represents the amount of money made from selling them.  Let's organize this information in a table:

Equation Type	[latex]x[/latex] represents	[latex]y[/latex] represents
Revenue Eqn.	number of frames	amount of money made selling frames
Cost Eqn.	number of frames	cost for making frames

Example

Given the same cost and revenue equations from the previous example, find the break-even point for the bike manufacturer.  Interpret the solution with words. Cost: [latex]y=0.85x+35,000[/latex] Revenue: [latex]y=1.55x[/latex]

Answer: Read and Understand: We want the break even point for this system that represents cost and revenue.  This means we want to find where the two lines cross, and we have learned a few different methods for doing this because this is the solution to the system of equations!  Substitution looks like the easiest method since the revenue equation is already solved for [latex]y[/latex], [latex]y=1.55x[/latex]. Define and Translate: Write the system of equations.

[latex]\begin{array}{l}\\ y=0.85x+35,000\hfill \\ y=1.55x\hfill \end{array}\\[/latex]

  Write and Solve: The equations are already written for us, so we just need to solve the system using substitution. Substitute the expression [latex]0.85x+35,000[/latex] from the first equation into the second equation and solve for [latex]x[/latex]. [latex-display]\begin{array}{r}0.85x+35,000=1.55x\\ 35,000=0.7x\,\,\,\\ 50,000=x\,\,\,\,\,\,\,\,\,\,\end{array}\\[/latex-display] Then, we substitute [latex]x=50,000[/latex] into either the cost equation or the revenue equation. [latex-display]1.55\left(50,000\right)=77,500\\[/latex-display] The break-even point is [latex]\left(50,000,77,500\right)[/latex].

Check and Interpret:

The solution to this system is[latex]\left(50,000,77,500\right)[/latex], but what does that mean? Think of a point as [latex]\left(x,y\right)[/latex], where in this case x is the quantity of bikes manufactured and [latex]y[/latex] is an amount of money. For our system [latex]y[/latex] represents two different things and [latex]x[/latex] represents one thing.  Refer to the table we made in the first example, shown below:

Equation Type	[latex]x[/latex] represents	[latex]y[/latex] represents
Revenue Eqn.	number of frames	amount of money made selling frames
Cost Eqn.	number of frames	cost for making frames

Let's interpret the solution with respect to the Cost equation first. [latex]x = 50,000[/latex] and [latex]y = 77,500[/latex].  Using our table, we can translate this as "the cost for producing [latex]50,000[/latex] bike frames is [latex]$75,500[/latex]". In the same way, the Revenue equation can be interpreted as "the amount of money the company makes from selling [latex]50,000[/latex] bike frames is [latex]$77,500[/latex]".