COMPARING A TABLE AND AN EQUATION

Let us assume that a function is written as equation and another function is represented by a table. To compare these two functions, we have to find the equation for the function represented by the table. 

Example  1 :

David and John buy MP3 files from different music services. The monthly cost, y dollars, for x songs is linear. The cost of David’s service is y = 0.50x + 10. The table given below shows the cost of John’s service. 

Based on the information given above, compare the services of David and John. 

Solution : 

Step 1 :

To compare the services of David and John, let us have both the services as equations. David's service is already given as equation. So, let us write John's service as equation. 

Choose any two ordered pairs from the table to find the slope.

Let us use the ordered pairs (5, 4.95) and (10, 9.90). 

Slope m  =  (y₂ - y₁) / (x₂ - x₁)

Slope m  =  (9.90 - 4.95) / (10 - 5)

Slope m  =  4.95 / 5

Slope m  =  0.99

Step 2 :

Find the y-intercept. Use the slope and any point.

Slope-intercept form equation of a line.

y  =  mx + b

Plug m  =  0.99, x  =  5 and y  =  4.95.

4.95  =  0.99 · 5 + b

4.95  =  4.95 + b 

Subtract 4.95 from both sides. 

0  =  b

Step 3 :

Substitute the slope and y-intercept.

y  =  0.99x + 0

y  =  0.99x

Step 4 :

To compare the services, find the cost in both the services when 30 songs are downloaded.

That is, find the values of "y" in the equations when x = 30. 

So, David's service is cheaper..

Example  2 :

Alex is choosing between buying books at the bookstore or buying online versions of the books for his tablet. The cost, y dollars, of ordering books online for x books is ya=a6.95xa+a1.50. The cost of buying the books at the bookstore is shown in the table. Which method of buying books is more expensive if Quentin wants to buy 6 books ?

Solution : 

Step 1 :

To know which method of buying books is more expensive if Quentin wants to buy 6 books, let us have both the methods as equations. Buying books online is already given as equation. So, let us write buying books at the store as equation. 

Choose any two ordered pairs from the table to find the slope.

Let us use the ordered pairs (1, 7.50) and (2, 15.00). 

Slope m  =  (y₂ - y₁) / (x₂ - x₁)

Slope m  =  (15.00 - 7.50) / (2 - 1)

Slope m  =  7.50 / 1

Slope m  =  7.50

Step 2 :

Find the y-intercept. Use the slope and any point.

Slope-intercept form equation of a line.

y  =  mx + b

Plug m  =  7.50, x  =  1 and y  =  7.50

7.50  =  7.50 · 1 + b

7.50  =  7.50 + b 

Subtract 7.50 from both sides. 

0  =  b

Step 3 :

Substitute the slope and y-intercept.

y  =  7.50x + 0

y  =  7.50x

Step 4 :

To know which method of buying books is more expensive if Quentin wants to buy 6 books, find the values of y in the equations when x = 6.

So, buying books at the book store is expensive. 

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