Please Help. System of Equation Problem

Question

At a store a frying pan costs two dollars less than two thirds as much as a food processor, and a coffee maker costs nine dollars less then three times as much as a frying pan. Ten food processors can be purchased for 111 dollars more than three times the sum of the costs of the other two items. How much does each item cost by itself?

Answer 1

Okay, so the tricky part here is that it's a word problem, which means we have to create the system of equations ourselves. My first step in this process is always to identify the variables.  In this case we have three different items (frying pan, food processor, coffee maker) and three separate statements about their relationships to each other.  So let's first assign variables to each of our three items.

 

x = frying pan

y = food processor

z = coffee maker

 

So far, so good.  Now, let's take each of those statements individually and translate them from English into math.  I do this using a process known as 'literal translation,' where you basically go one word at a time through the sentence and convert it all into math symbols.  Here's the first statement:

 

a frying pan costs two dollars less than two thirds as much as a food processor

 

Okay, 'less than' means you subtract, but because of the way English syntax works compared to math syntax, we'll actually want to put the 'minus 2' at the end of the math expression.

 

x =                      - 2

 

'two-thirds as much' literally just means the fraction 2/3, multiplied by the following quantity, which in this case is the variable y.

 

so

 

x = (2/3)y - 2

 

You'll repeat this process with the other two statements to create two more equations:

 

a coffee maker costs nine dollars less then three times as much as a frying pan.

 

z = 3x - 9

 

and finally,

 

(Ten food processors) (can be purchased for) (111 dollars more than) (three times) (the sum of the costs of the other two items)

 

10y = 111 + 3(x + z)

 

So there are your three equations for the system.  You should now be able to solve the system for all three variables from there.