Tutorial

Solving Systems of Equations

 

Part I: The tutorial. 

 

You decided to open a donut shop renting a store in town. Your cost to produce one donut is $0.05, with an upfront expense for rent, labor and equipment is $500. If you sell each donut for $0.35, how many donuts will you have to sell until you start making money?

 

1. Define your variables:

Let x = the number of donuts sold.

                              y = amount of money

 

2. Write the two equations:

                      (1)  Cost = .05x + 500

                      (2)  Sales Income = .35x

          Find where Cost = Sales Income.

 

We know once our sales income is greater than our cost for producing and selling the donuts, we will start making money. How can we find the point where we break-even?

 

3. Find the Break-Even point, which is the same as the intersection point:

 

a.      Graphing:

i.         Input the two equations in your graphing calculator and graph using the given window.

 

                              

 

ii.        Find intersection point using the Intersect option under    TRACE.

 

 

The Break-Even Point is (1666.67, 583,33) which means that you will start making money if you sell 1667 donuts since at this point your cost will be equal to your sales income.

 

 

 

b.  Matrices:

i.  Input the following values in a 2 x 3 matrix: (Note: you need to          

     write the equations in standard form)

 

            

 

      ii.  Then perform rref:

           Matrix ... Math ... B:rref( ... enter the Matrix  A) ... then press   

              ENTER

           to find the values for x and y:

 

 

Your values for x and y are 1666.66 and 583.33, respectively.

Hence, the results are identical to the results above.

 

Note:  If you have systems of linear equations with three variables you will not be able to graph them using your TI-83+. You then will have to resort to the matrix method.