# Thread: System of Equations - Application Problem

1. ## System of Equations - Application Problem

Each school district in Texas has a budget by which to operate. School administrators plan spending so that the exprenses for a school are less than the revenue. The greatest yearly exprense for a school is payroll for teachers and other staff. The cost of building new schools, buying school buses, or other long-term expenses are called capital expenses. Schools may also have to pay out money for loans or to people who have bought bonds, which support the schools. These are called debt service expenses.

The amout of revenue, $R$, per student to Texas schools for the 1999-2000 school year to ther 2003-2004 school year can be approximated by

Revenue: $R = 5732(1.025)^t$

where $t$ represents the school year with $t = 0$ corresponding to the 1999-2000 school year. For the same time period, expenses per student for Texas schools can be represented by the following formulas:

Payroll expenses: $P = 115t + 4598$
Capital expenses: $C = 18t + 191$
Debt service expenses: $D = 2t^2 + 8t + 37$
Miscellanueous expenses: $M = 19t + 894$

1. Find an equation that represents the total expenses, $E$, per student.

• For this one...would you just add up all the systems of equations above to get the equation that represents the total expenses per student?

1. Graph the models for the revenue per student and the total expenses per student on the same set of coordinate axes.

• I'll graph this...but I just want to make sure I got the right equation representing total expenses per student in the previous question.

1. In which school year is the revenue per student approximately equal to the total expenses per student? Explain.

• Like previous statement before.

1. In which school year is the difference between the revenue per student and the expenses per student the greatest? What is the difference?

• Like previous statement before.

1. In which school year is the percentage that capital expenses are of total expenses the greatest? What is the percentage?

• Like previous statement before.

2. 1) I would agree: add all the different categories of expenses to get an expression for the total expense.

2) Graph the "revenue" formula they gave you and the 'expenses" formula you just came up with.

3) Eye-ball the graph (or use the "intersection" utility on your graphing calculator) to approximate the x-value (okay, t-value) for which the two lines cross.

4) Eye-ball the graph, and locate the x-value for which the two lines are the farthest apart.

5) Divide the expression for "C" by the expression for "E" to get an expression for the percentage that C is of E. Graph the result. Eye-ball to find the x-value for which the graph is lowest. The y-value for that point will be the percentage (as a decimal, of course; you'll need to convert to a percentage).

3. Originally Posted by stapel
1) I would agree: add all the different categories of expenses to get an expression for the total expense.

2) Graph the "revenue" formula they gave you and the 'expenses" formula you just came up with.

3) Eye-ball the graph (or use the "intersection" utility on your graphing calculator) to approximate the x-value (okay, t-value) for which the two lines cross.

4) Eye-ball the graph, and locate the x-value for which the two lines are the farthest apart.

5) Divide the expression for "C" by the expression for "E" to get an expression for the percentage that C is of E. Graph the result. Eye-ball to find the x-value for which the graph is lowest. The y-value for that point will be the percentage (as a decimal, of course; you'll need to convert to a percentage).

Alright...thanks alot. I just wanted to make sure I had to add up all the equations to get the equation for the total expense.