Pearl writes down seven consecutive integers, and adds them up. The sum of the integers is equal to 14 times the largest of the seven integers. What is the smallest integer that Pearl wrote down?
\(\phantom{14/3 times}\)
+2666 We have the equation: \(x + x + 1 + x + 2 + x + \cdot \cdot \cdot+ x+6 = 14(x+6)\)
Simplify to: \(7x + 1 + 2 +3 + \cdot \cdot \cdot + 6 = 14x + 84\)
Furthur simplify the left-hand side: \(7x + 21 = 14x+84\)
Subtract 7x from both sides: \(21 = 7x + 84\)
Subtract 84 from both sides: \(7x = -63\)
Divide by 7: \(x = \color{brown}\boxed{-9}\)
