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Writing expressions

The average # of tea drinkers in the year 2000 was 12.5 million and the average # of coffee drinkers was 16 million.  During the next 3 years, the # of tea drinkers would increase an average of 5 million per year and the number of coffee drinkers would decrease an average of 2 million per year.  Write an expression describing the relationship between the # of tea drinkers and time (x years after 2000), and an expression describing the # of coffee drinkers and time.

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3 Answers

Tea: f(x) = 12.5 + 5x

Coffee: f(x) = 16 - 2x

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Attn: Slope = rate here. Increasing leads to positive slope, and decreasing leads to negative slope.

Comments

You have a typo. Should be 16 - 2x for coffee.

Comment

Lets start with our givens.

Our year is 2000 which for our purposes is the starting point so we can just call it x=0, where x is the amount of years after 2000 (as is asked of us in the problem, x will stand for years). The # of tea drinkers at this time is 12.5 million and the # of coffee drinkers is 16 million. We are told that the # of tea drinkers is increased by 5 million and the # of coffee drinkers decreased by 2 million each year after the year 2000 (x=0).

So now we need to make an expression involving x for tea drinkers (I won't do coffee drinkers because the process is quite similar and it's no fun to just give away all the anwers). Well if we think about this a bit more, for every year that passes, in other words each time x increases by 1, we have an increase of 5 million drinkers. We have now the following progression:

x=0; no increase

x=1; +5 million

x=2; +5 million +5 million = +10 million

x=3; +5 mil +5 mil +5 mil = +15 million...and so on.

As we can see, for any arbitrary year we can say that the amount of tea drinkers increases by the following expression: 5mil*x or for simplicity 5x.

Given now our change for every year after 2000 we also need to include the year 2000 in our expression. To do this all we need to do is have a starting value of 12.5 that won't be affected by x. Now our expression will look like this:

#T = 5x + 12.5 (Answer)

To test this we plug in x=0 (year 2000) and surely enough we get our starting value at 2000 as 12.5 million just as we needed.

Again, this process is done for coffee drinkers as well except this time we will have a decrease for every x that will be:

x=1; -2 mil

x=2; -2 mil -2 mil = -4 mil....

The rest is up to you!

Hope this helped.

Let's use T for the tea and C for the coffee, measured in millions. Then their relation with time is:

T(x) = 12.5 + 5x and C(x) = 16 - 2x

If you also wanted to relate the two with each other, you can eliminate x:

2T + 5C = (25 + 10x) + (80 - 10x)

2T + 5C = 105.

 

Comments

I'm pretty sure it would be best to show the student how you arrived at 2T+5C instead of just stating it. A better approach would be to solve for x in terms of either T or C and then plug it in to the other expression, at least it would be easier for the student to understand that.

Yes. I just used the least common multiple of the coefficients of x terms, which is 10. Multiplying T(x) by 2 make's it's x-coefficient 10, and multiplying C(x) by 5 makes it's x-coefficient -10. Now adding the resulting equations makes the x-terms cancel.

Comment