An envelope has an area of 50 square inches and a perimeter of 30 inches. What are the dimensions of the envelope? _____ inches by _____ inches

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An envelope has an area of 50 square inches and a perimeter of 30 inches. What are the dimensions of the envelope?  _____ inches by _____ inches

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Define Variables: Let's denote the length of the envelope as l and the width as w. The area A of a rectangle is given by the formula A = l * w, and the perimeter P is given by P = 2l + 2w.Area and Perimeter Equations: We are given the area of the envelope as 50 square inches, so we have the equation l * w = 50.Solve System of Equations: We are also given the perimeter of the envelope as 30 inches, so we have the equation 2l + 2w = 30. We can simplify this equation by dividing both sides by 2, which gives us l + w = 15.Quadratic Equation Simplification: Now we have a system of two equations: 1. l * w = 50 2. l + w = 15  We can solve this system by expressing one variable in terms of the other using the second equation. Let's solve for w: w = 15 - l.Factor Quadratic Equation: Substitute w from the second equation into the first equation: l * (15 - l) = 50 This simplifies to a quadratic equation: l^2 - 15l + 50 = 0Find Possible Solutions: To solve the quadratic equation, we can factor it: (l - 10)(l - 5) = 0 This gives us two possible solutions for l: l = 10 or l = 5.If l = 10, then w = 15 - l = 15 - 10 = 5. If l = 5, then w = 15 - l = 15 - 5 = 10. Both pairs (l = 10, w = 5) and (l = 5, w = 10) are valid solutions because they are interchangeable due to the envelope's length and width being relative terms.

An envelope has an area of $50$ square inches and a perimeter of $30$ inches. What are the dimensions of the envelope? $\newline$ $\_\_\_$ inches by $\_\_\_$ inches

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An envelope has an area of $50$ square inches and a perimeter of $30$ inches. What are the dimensions of the envelope? $\newline$ $\_\_\_$ inches by $\_\_\_$ inches