Solve the following word problem: "Twice the sum of a number and three is equal to triple that number. Triple a number is the same as double the sum of the number and three. What is the number?"
Let x represent the unknown number. Write an equation that reflects the mathematical information. Twice the sum of a number and three is written mathematically as 2(x + 3) and is set to equal triple that number, which is written as 3x. The polynomial equation is 2(x + 3) = 3x.
Distribute 2 throughout the parenthetical terms to simplify that side of the equation. 2 x x = 2x, and 2 x 3 = 6. The simplified equation is 2x + 6 = 3x.
Move 2x to the other side of the equation by using the opposite mathematical property. The goal is to get x by itself on one side of the equation so that it is set to equal a value.
Subtract 2x from both sides of the equation. Use subtraction, because 2x is positive, and subtract it from both sides to keep the equation balanced. 2x -- 2x + 6 = 3x -- 2x.
Simplify the equation to 6 = x. Substitute the value of x back into the equation to double check your work. 2(6 + 3) = 3(6). Simplify that to 2(9) = 18 = 3(6) = 18 = 18.
Solve the following word problem: "A 22-foot board is cut into two pieces. The second piece is 2 foot shorter than twice the length of the first. What is the length of both pieces?"
Write a formula that matches the information in the word problem: 22 = L + (2L -- 2). L represents the first piece, and (2L -- 2) represents the second piece . The first piece, L, and the second piece, (2L -- 2), equal 22.
Combine like terms: L + 2L = 3L. Simplify: 22 = 3L -- 2.
Begin isolating the variable by adding 2 to both sides of the equation: 22 + 2 = 3L -- 2 + 2. Simplify: 24 = 3L.
Divide both sides of the equation by 3 to isolate x: 24 ÷ 3 = 3L ÷ 3. Simplify: 8 = L. This is the length of the first piece.
Substitute the value of L back into the polynomial equation: 22 = 8 + (2(8) -- 2). Solve the terms in parentheses to find the length of the second piece: 2(8) = 16 -- 2 = 14. The formula with both measurements included is 22 = 8 + 14, which when simplified is 22 = 22. verifying the solution.