Can you help me find the linear function for this story problem?

Question

A teacher decides to scale her class's grades upwards. She raises the lowest score, a 47, to a 65. She raises the highest score, a 78, to a 90. What's the linear function that would fairly convert the other test scores?

I do know the answer to this problem, but what I don't know is how to get there. What steps should I take?

Answer 1

Let x to be the old score, and let y to be the new score. So we have a linear function:
y = ax + b
with a and b the parameters to be determined.
Based on the given data, we have:
65 = 47a + b (1)
90 = 78a + b (2)
Do subtraction (2) - (1), we get:
25 = 31a
So a = 25/31 and thus b = 90 - 78*25/31 = 27+3/31
Hence the required linear function is:
y = (25/31)x + (27+3/31)