Undoing a Conversion

Part of the Functions & Transformations section of Coddy's Algebra 2 journey. Lesson 52 of 58.

A conversion rule turns one unit into another, and its inverse turns it back. In the US, a man's shoe size is about 3 times his foot length in inches, minus 22: f(x) = 3x - 22. A 10 inch foot takes size 8. To go from a size back to a length, you need f^-1(x).

The method is the one from the inverse chapter. Write the rule as y = 3x - 22, swap x and y, then solve for y. Swapping gives x = 3y - 22. Add 22 to get x + 22 = 3y, then divide by 3: y = (x + 22)/3. Check: size 8 gives 30 over 3, which is 10 inches, the length we started from.

Why swap? The inverse takes the old output as its input, and we keep the letter x for an input. When the rule multiplies by a fraction, divide by that fraction, which is the same as multiplying by its flip: x - 4 = 2/3 y gives y = 3/2(x - 4).

Challenge

The rule F = 9/5 C + 32 turns a temperature in Celsius into Fahrenheit. To undo it, the board holds the rule with the letters swapped: x = 9/5 y + 32, where x is now the Fahrenheit reading. Solve for y to get the rule that turns Fahrenheit back into Celsius.

Try it yourself

x = 9/5 y + 32

Solve for y