Let $f$ be a linear function for which $f(6) - f(2) = 12.$ What is $f(12) - f(2)?$
Since $f$ is linear, it has the form $f(x) = ax + b$ for some constants $a$ and $b.$ Then $f(6) - f(2) = 12$ means that $a(6) + b - (a(2) + b) = 12.$ Simplifying, I get $4a = 12,$ so $a = 3.$ Now I want to find $f(12) - f(2) = a(12) + b - (a(2) + b).$ Substituting $a = 3,$ I get $3(12) + b - (3(2) + b) = 36 - 6 = 30.$ 