let 'x' and 'y' be the lengths of each piece and let x>y

a carpenter saws a board that is 22 feet long into two pieces

x + y = 22

one piece is to be 1 foot longer than twice the length of the shorter piece

x = 1 + 2y

by solving the system of equations

x + y = 22

x = 1 + 2y

we find

x = 15 ft

y = 7 ft

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the lengths of the two pieces are 15 ft and 7 ft.