Working Backward

Sometimes the story gives the output and asks for the input. A student paid the music school 135 dollars in all. How many lessons did they take? That asks for the n with $C\left(n\right) = 135$.

Use the rule to turn it into an equation: $25+10n = 135$. Undo the fixed part first, since it was added last when the cost was built: subtract 25 to get $10n = 110$. Then undo the multiplication: divide by 10, so $n = 11$. Eleven lessons.

Check it forwards: 25 plus 10 times 11 is 135. And check it against the story: the answer counts lessons, so it must be a whole number. If you ever get 11.5, either the working or the bill is wrong.