Algebra word problem

Question

The cost of three notebooks and four pencils is $8.50. The cost of five notebooks and eight pencils is $14.50. Determine the cost of one notebook and the cost of one pencil. 

Answer 1

As are most algebra word problems, this is a system of linear equations problem. Any time you have a rate, you have a linear equation waiting to be written. In this case, we are not given rates outrightly but are given two linear equations.

 

Let n = cost of 1 notebook

Let p = cost of 1 pencil

 

Then,

 

3n + 4p = 8.5

5n + 8p = 14.5

 

You can solve for one variable in terms of the other and then substitute into the remaining equation. However, I prefer (Gaussian) elimination.

 

Set up an addition of the two equations:

    3n + 4p = 8.5
+  5n + 8p = 14.5

 

Multiply the top equation by -2 so that the p-containing terms cancel each other out:

 

    -2(3n + 4p = 8.5)
+      5n + 8p = 14.5

         -n +  0  = -2.5

 

So after dividing both sides by -1, we see that n = $2.5. Plugging into the first equation gives p = $0.25.

 

 

 

 

 

 

 

 

 

3n + 4p = 8.5

5n + 8p = 14.5

Answer 2

Hello Karley,

This is a problem with two equations and two unknown variables, the cost of the notebook and the cost of the pencil.

First, it helps to assign letter variables to the costs and write out the equations. I will assign "x" as the cost of one notebook and "y" as the cost of one pencil:

 

3x + 4y = $8.50

5x + 8y = $14.50

 

From here, you can proceed to the solution in one of four ways. It doesn't matter which one you choose, they will all lead to the correct answer. I'll solve the top equation for x and insert it into the bottom equation:

 

Eq. 1: x = ($8.50 - 4y)/3

Eq. 2: 5($8.50 - 4y)/3 + 8y = $14.50

 

You can now use Eq. 2 to solve for y, what I have assigned as the cost of one pencil:

 

(5/3)($8.50) - (20/3)y + 8y= $14.50

After rearranging and solving for y:

y = -(3/4)((5/3)($8.50) - $14.50)

= -$10.625 + $10.875

= $0.25 / pencil

 

Plug this value in for y in the top equation to determine x, the cost of one notebook:

 

3x + 4($0.25) = $8.50

3x = $7.50

x = $2.50 / notebook

 

To double check your answer, plug values in to both equations and see if they are satisfied. I am sorry about the ugly parentheses everywhere. "Bookkeeping" is very important in algebra problems, and math in general really.

 

Best,

- Avery