SOLUTION: My text book is asking a question that I have no idea how to answer. It says consumption, (c) is to be linear and that it is a function of (x), income. It is saying to write the

Algebra ->  Expressions -> SOLUTION: My text book is asking a question that I have no idea how to answer. It says consumption, (c) is to be linear and that it is a function of (x), income. It is saying to write the       Log On


   

Question 103953: My text book is asking a question that I have no idea how to answer. It says consumption, (c) is to be linear and that it is a function of (x), income. It is saying to write the consumption function. Here is the information provided:
Consumption (c) = $18 million
Income (x) = $12 million
&
Consumption (c) = $24 million
Income (x) = $18 million
Any thoughts on this , thanks.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Think of c as y to get one pair of values: x=12 million and y=18 million
and another pair: x=18 million and y=24 million


So we basically can think of these pairs as points on a coordinate system
(12,18) and (18,24) (notice I've taken out the "millions". This will simplify things if we remember everything is in millions and not write it every time)


So let's find the equation of the line through the points (12,18) and (18,24)



First lets find the slope through the points (12,18) and (18,24)

m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula (note: is the first point (12,18) and is the second point (18,24))

m=%2824-18%29%2F%2818-12%29 Plug in y%5B2%5D=24,y%5B1%5D=18,x%5B2%5D=18,x%5B1%5D=12 (these are the coordinates of given points)

m=+6%2F6 Subtract the terms in the numerator 24-18 to get 6. Subtract the terms in the denominator 18-12 to get 6


m=1 Reduce

So the slope is
m=1

------------------------------------------------


Now let's use the point-slope formula to find the equation of the line:



------Point-Slope Formula------
y-y%5B1%5D=m%28x-x%5B1%5D%29 where m is the slope, and is one of the given points

So lets use the Point-Slope Formula to find the equation of the line

y-18=%281%29%28x-12%29 Plug in m=1, x%5B1%5D=12, and y%5B1%5D=18 (these values are given)


y-18=1x%2B%281%29%28-12%29 Distribute 1

y-18=1x-12 Multiply 1 and -12 to get -12

y=1x-12%2B18 Add 18 to both sides to isolate y

y=1x%2B6 Combine like terms -12 and 18 to get 6
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Answer:


So the equation of the line which goes through the points (12,18) and (18,24) is:y=1x%2B6

The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=1 and the y-intercept is b=6

Notice if we graph the equation y=1x%2B6 and plot the points (12,18) and (18,24), we get this: (note: if you need help with graphing, check out this solver)

Graph of y=1x%2B6 through the points (12,18) and (18,24)

Notice how the two points lie on the line. This graphically verifies our answer.


So to make a function out of y=x%2B6, simply replace y with C%28x%29 to get


C%28x%29=x%2B6