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**Formulas and Problem Solving**

§ 2.5 Formulas and Problem Solving

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Formulas A formula is an equation that states a known relationship among multiple quantities (has more than one variable in it) A = lw (Area of a rectangle = length · width) I = PRT (Simple Interest = Principal · Rate · Time) P = a + b + c (Perimeter of a triangle = side a + side b + side c) d = rt (distance = rate · time) V = lwh (Volume of a rectangular solid = length · width · height)

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**Using Formulas Example:**

A flower bed is in the shape of a triangle with one side twice the length of the shortest side, and the third side is 30 feet more than the length of the shortest side. Find the dimensions if the perimeter is 102 feet. 1.) UNDERSTAND Read and reread the problem. Recall that the formula for the perimeter of a triangle is P = a + b + c. If we let x = the length of the shortest side, then 2x = the length of the second side, and x + 30 = the length of the third side Continued

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**Using Formulas Example continued: 2.) TRANSLATE Formula: P = a + b + c**

Substitute: = x + 2x + x + 30 3.) SOLVE 102 = x + 2x + x + 30 102 = 4x Simplify right side. 102 – 30 = 4x + 30 – Subtract 30 from both sides. 72 = 4x Simplify both sides. Divide both sides by 4. 18 = x Simplify both sides. Continued

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**Using Formulas Example continued: 4.) INTERPRET**

Check: If the shortest side of the triangle is 18 feet, then the second side is 2(18) = 36 feet, and the third side is = 48 feet. This gives a perimeter of P = = 102 feet, the correct perimeter. State: The three sides of the triangle have a length of 18 feet, 36 feet, and 48 feet.

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**Solving Formulas Solving Equations for a Specific Variable**

It is often necessary to rewrite a formula so that it is solved for one of the variables. This is accomplished by isolating the designated variable on one side of the equal sign. Solving Equations for a Specific Variable Multiply to clear fractions. Use distributive property to remove grouping symbols. Combine like terms to simplify each side. Get all terms containing specified variable on the same side, other terms on the opposite side. Isolate the specified variable.

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**Solving Equations for a Specific Variable**

Example: Solve for n. Divide both sides by mr. Simplify right side.

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**Solving Equations for a Specific Variable**

Example: Solve for T. Subtract P from both sides. Simplify right side. Divide both sides by PR. Simplify right side.

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**Solving Equations for a Specific Variable**

Example: Solve for P. Factor out P from both terms on the right side. Divide both sides by 1 + RT. Simplify the right side.

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