Two containers are being filled with water. one begins with 8 gallons of water and is filled at a rate of 3.5 gallons per minute. the other begins with 24 gallons and is filled at 3.25 gallons per minute.
part a. write an equation that represents the amount of water w, in gallons, with respect to time t, in minutes for each container.
part b. solve the system of equations. show your work.
part c. how long would it take for both of the containers to have the same amount of water? how much water would be in each container?

A.

B. see below

C. 64 minutes; 232 gallons of water

A. The form of linear equation that makes use of an initial value (b) and a rate of change (m) is ...

  y = mx + b . . . . . "slope-intercept" form

Here, your independent variable is t, not x. Your dependent variable is w, not y. Given an initial value of 8 gallons and a rate of change of 3.5 gallons per minute, the first equation can be written ...

  w = 3.5t + 8

In similar fashion, the second equation can be written ...

  w = 3.25t + 24

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B. There are a variety of ways these equations can be solved to find w and t that satisfy both. One way is by using a graphing calculator (see attached). You can also use substitution or elimination. Here, it is convenient to simply subtract the second equation from the first:

  (w) -(w) = (3.5t +8) -(3.25t +24)

  0 = 0.25t -16 . . . . . . simplify

  0 = t - 64 . . . . . . . . . multiply by 4

  64 = t . . . . . . . . . . . . add 64

Now, we can find the corresponding value of w using either equation. Using the first, we have ...

  w = 3.5·64 +8 = 224 +8 = 232

The solution is (t, w) = (64, 232).

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C. The solution of part B tells us that t=64 minutes, and w=232 gallons.

It will take 64 minutes for both containers to have the same amount of water, which will be 232 gallons.