Word problem equation help

Question

When a city had 21000 inhabitants, there were 250 businesses operating there. For every new 150 residents, one more business opened. Write an equation that relates the number of people in the city to the number of businesses.

 

I just need help understanding which numbers to use as my variables and such?

Answer 1

Hi Carl,

 

Let's begin by determining our dependent (output) variable (y) and our independent (input) variable (x).

We are told to write an equation relating the number of residents to the number of businesses.  This means that given an input (number of new residents) will result in an output (number of new businesses):

    y (number of businesses) - output variable

    x (number of residents) - input variable

 

Now let's see what kind of relationship exists between our variables.  We are told for every 150 new residents (input) there will be 1 new business (output).  This describes a linear relationship (a uniform increment to both variables).

 

Since we now know that we have a linear relationship, we also know that our equation will be the equation of a line:

    y = mx + b (equation of a line)

 

We need to know two pieces of information to write an equation of a line:

    1.  slope (m) = (change in y) / (change in x)

    2.  y-intercept (b) = value of y when x = 0 

 

We already have the slope of the line given to us in the problem:

    change in y (number of businesses) = 1

    change in x (number of residents) = 150

    m = (change in y) / (change in x)

    m = 1/150

 

To calculate the value of b (y-intercept), we need to plug these values into the equation of a line:

    1.  The slope (m) - our calculated value of 1/150.

    2.  The coordinates (x,y) of a point on the line,

         which we have from our problem:

             (a)  When there were 21000 inhabitants (x)

                   then there were 250 business (y):

 

Plugging in these values into equation of a line and solving for b:

    y = mx + b

    250 = (1/150)(21000) + b

    250 = 140 + b

    b = 250 - 140

    b = 110

 

Putting all the pieces of information together, we have our equation:

    y = mx + b

    m = 1/150

    b = 110

    y = (1/150)x + 110 (our equation)

 

Once we determined what were our input and output variables and the kind of relationship existing between these variables, it allowed us to write an equation expressing that relationship.

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan.

Answer 2

Let P = number of people

Let N = number of businesses 

 

I don't know what level of math you are in, but if you are familiar with the slope-intercept form of a line (y = mx + b) you should be able to recognize that this will be a linear equation with a slope of 150 (meaning that every time a new business opens there are 150 new residents).  So your equation will have the form  

 

P = 150*N + b  

 

You can use the given information (21000 inhabitants when there were 250 businesses) to figure out the y-intercept  

 

21000 = 150*(250) + b

21000 = 37500 + b

b = -16500  

 

So the equation is  P = 150*N - 16500  

 

You can check the answer by noting that every time N goes up by 1, P will increase by 150;  and plugging in 250 for N will give P = 150*250 - 16500 = 37500 - 16500 = 21000 inhabitants.