Cut a 25-foot board into three pieces. The first piece should be 2 feet more than twice the length of the second piece and the third piece should be 3 feet longer than the second piece. Find the length of all three pieces.
Write a polynomial equation with the information. Since the other lengths are dependent on the length of the second piece, use the variable x to represent the second piece, 2x + 2 to represent the first piece and x + 3 to represent the third. The length of the board is 25 feet, so set the total length equal to the polynomials: 25 = x + (2x + 2) + (x + 3).
Remove all the parentheses: 25 = x + 2x + 2 + x + 3. Combine all the like terms: 25 = (x + 2x + x) + (2 + 3) = 25 = 4x + 5.
Subtract 5 from both sides of the equation: 25 -- 5 = 4x + 5 -- 5 = 20 = 4x. Use subtraction because 5 is originally added; do it to both sides to keep the equation balanced. This step not only combines like terms, but also isolates the variable on one side of the equation.
Divide both sides by the coefficient of the variable: 20 ÷ 4 = 4x ÷ 4. Simplify to 5 = x. This is the length of the second piece.
Find the length of the first piece and the third piece by plugging the value of x back into the formulas: (2x + 2) = (2 x 5) + 2 = 12. Therefore, the first piece is 12 feet long and (x + 3) = (5 + 3) = 8. Therefore, the third piece is 8 feet long.
Rewrite the formula with the information to check for equality: 25 = 12 + 5 + 8, which is a true statement.