# A business had annual retail sales of $224,000 in 1989 and $186,500 in 1992. Assume the annual decrease in sales followes a linear pattern. Write a linear equation giving sales S in terms of the...

A business had annual retail sales of $224,000 in 1989 and $186,500 in 1992. Assume the annual decrease in sales followes a linear pattern.

Write a linear equation giving sales S in terms of the year t where t=0 corresponds to 1989.

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### 1 Answer

Hi, 4214,

I hope this helps. Given this problem, I assume your class has written the linear equations of problems (y=mx+b) given 2 points, a point and slope, etc. Well, in these word problems, those conditions are given. We just need to get them out. Like in this one.

The first thing to make sure of is the year.

t = 0 for year 1989, so

t = 1 for year 1990

t = 2 for year 1991

t = 3 for year 1992, etc.

This is commonly done for years in problems. The year t would be x.

So, for the given material:

$224,000 in 1989

The dollars must be y. So, with the year x and dollars y, we have:

(0, 224,000) for t = 5

For the next one, we have $186,500 in 1992, or:

(3,186.500)

So, now, we have our two points. We need m, the slope, and b, the y intercept, for the equation. For the y intercept, all y intercepts have x = 0, as in (0,y) for all y intercepts. Well, one of our points have x = 0, so that must be the y intercept, (0, 224,000). For the equation, we only use 224,000. That is b.

For the slope, we can do a couple of things. Using the definite of slope:

m - (y2 - y1)/(x2 - x1)

And, plugging in the x's and y's:

m = (186,500 - 224,000)/(3 - 0)

So, m = -12,500

And, remember, we have b = 224,000 and m = -12,500, then the equation is:

y = -12,500x + 224,000

God luck, 4214. I hope this helps.

Till Then,

Steve