**Writing Expressions Involving Rate of Change**

These real-world problems can be best translated when broken down into their components (variables and operations). When you see the words “is” or “are”, this is the points where you set-up the equality. Whenever you see the word “per”, “each” the implication
is a multiplication. This indicates the rate of change between the variables.

The general format for these problems is:

Dependent Variable = Fixed Value + Rate of Change * Independent Variable.

The fixed value is generally a fixed value which does not change. Most commonly, it will be the initial value in a situation.

__Example 1:__

“Mark is purchasing a new computer. The cost of the computer is $2400 after tax. He will make monthly payments of $150. Write an equation which describes the balance on the account after any given number of months”

Variables present: balance and number of months.

The rate of change in this case is the $150 per month. The word “per” is the indicator of the rate of change.

The relationship between the variables is:

Dependent Variable = Fixed Value + Rate of Change * Independent Variable.

“Balance” = “Cost of Computer” – “150” per “month”

b = 2400 – 150*m.

Because the balance on the account is going to be lower as the time passes, you will subtract from the initial value.

__Example 2:__

“Mr. Fellow bought a refrigerator that cost $1200 including tax. The cost of electricity to run the refrigerator is estimated at $63 per year. Write an equation which represents the total cost of operation”.

Relationship variables are: total cost of operation and number of years.

Dependent Variable = Fixed Value + Rate of Change * Independent Variable.

Total cost of operation = initial cost + “cost” per “year”

t = 1200 + 63*y

__Example 3:__

“Vicki works as a sales associate in a department store. She earns $6 per hour, plus a commission of 3% on her sales. Write an equation which describes her total earnings”.

Aside from numbers, the relationship can be described as “Total earnings depend on sales and number of hours worked”. There are three variables. In this particular case, each variable carries its own rate of change and there is no fixed value as her earnings
start at $0.

Setting-up the relationship:

Dependent Variable = Rate*Independent Variable + Rate*Independent Variable

“Total Earnings” = “$6” per “Hour” + “3%” per “sales (in dollars).”

e = 6*h + 0.03*s.

__Example 4:__

“Passengers on a commercial flight are able to make in-flight calls using the built-in telephone system. The calls cost $3 to connect plus $1.85 each minutes. Write an equation the represents the total cost t, to make a call which lasts n number of minutes.”

Dependent Variable = Fixed Value + Rate of Change * Independent Variable.

“Total Cost” = “Cost to connect” + “cost” “per” “minute”

t = 3 + 1.85*m