SOLUTION: Water boils as C (degrees Celsius) which is F (degrees Fahrenheit) and freezes at C which is F. Let x denote the temperature of an object in F and y the temperature of the sam

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Question 612782: Water boils as C (degrees Celsius) which is F (degrees Fahrenheit) and freezes at C which is F. Let x denote the temperature of an object in F and y the temperature of the same object in C.
(a) Express x in terms of y.
x(y)= F
(b) Express y in terms of x.
y(x)= C
(c) Use (b) to convert F into Celsius.
C
(d) Use (a) to convert your answer to (c) back into Fahrenheit.
F
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Water boils at 100%5EoC (degrees Celsius) which is 212%5EoF (degrees Fahrenheit) and freezes at 0%5EoC which is 32%5EoF. Let x denote the temperature of an object in F and y the temperature of the same object in C.
So we have two points (A and B) of the line (the linear function) that relates y and x.
or
With the coordinates of two points, you can determine the slope and the equation of the line.

(a) Express x in terms of y.
x(y)= F
In the graph on the left, x increases by 212%5EoF-32%5EoF=180%5EoF (the vertical rise from A to B),
while y increases by 100%5EoC-0%5EoC=100%5EoC (the horizontal run from A to B).
So the slope is %28212-32%29%2F%28100-0%29=180%2F100=9%2F5
With that slope, and the y-intercept 32%5EoF from point A, we can write the slope-intercept form of the equation as
highlight%28x%28y%29=%289%2F5%29%2Ay%2B32%29F

(b) Express y in terms of x.
y(x)= C
You start from scratch calculating the slope of the line, and then using that slope and point A to get the point slope form of the equation.
However, the easiest way to get the needed equation is to solve for y in the equation found in part (a).
x=%289%2F5%29%2Ay%2B32 --> x-32=%289%2F5%29%2Ay --> %285%2F9%29%28x-32%29=%285%2F9%29%289%2F5%29%2Ay --> %285%2F9%29%28x-32%29=y
I would stop at highlight%28y%28x%29=%285%2F9%29%2A%28x-32%29%29C, because that is the equation I would use to do mental math conversions.
Transforming that into the slope-intercept form would give you a less user-friendly equation:
y%28x%29=%285%2F9%29%2A%28x-32%29 --> x%28y%29=%285%2F9%29%2Ay-%285%2F9%29%2A32 --> x%28y%29=%285%2F9%29%2Ay-160%2F9

(c) Use (b) to convert F into Celsius.
If I wanted to convert 105%5EoF into degrees Celsius to tell a very ill tourist his/her body temperature in familiar units, I would use the equation from part (b), to get
y%28105%29=%285%2F9%29%2A%28105-32%29=%285%2F9%29%2A73=40.6C (rounded from 40.55555....)

(d) Use (a) to convert your answer to (c) back into Fahrenheit.
x%2840.6%29=%289%2F5%29%2A40.6%2B32%29=105.1F (rounded from 105.08)
Yes, rounding errors caused the final answer to be a tiny bit different from the 105%5EoF I started with.