Students know that the same warmth registers differently on the Celsius and the Fahrenheit scales. They might be surprised to know that there is a point at which they are the same. That temperature could be found by using a system of equations. Celsius and Fahrenheit are related with the equation F=(9/5)C+32. Since we want to find out when the two scales are the same, we also need the equation F=C.
Initially, students might make up a table of values and notice that when it is warmer, temperature readings are farther apart -
100 degrees Celsius = 212 degrees Fahrenheit
40 degrees Celsius = 104 degrees Fahrenheit
0 degrees Celsius = 32 degrees Fahrenheit
The differences are getting closer together - 112 degrees apart, then 64, then 32.
If we change the variables to x's and y's, the equations would perhaps look more familiar to students:
y=(9/5)x+32 and y=1x+0
In this form students should see these are both linear equations, but have different slopes. While they might not know where it is yet, clearly those lines must have an intersection someplace. So, being straight lines, there must be one and only one temperature that is the same for each scale.
To get that temperature, the substitution method is probably the easiest way to go.
x=(9/5)x+32
5x=9x+160
-4x=160
x=-40
And so if x is -40, y must also be -40.
Being linear equations and already in slope-intercept form, algebra students should be able to easily graph them. Doing so, they would find an intersection point (or at least fairly close) of (-40, -40).
