1. Introduction The formula for finding the area of a trapezoid, is a formula because it involves two or more variables that have specific and special relationships with the other variables. In this case, the formula involves four variables: b 1, a base of the trapezoid; b 2, the other base of the trapezoid; h, the height of the trapezoid; and A, the area of the trapezoid. 1 5.3.4: Rearranging Formulas

2. Introduction When you rearrange the formula, you are simply changing the focus of the formula. In its given form, the focus is A, the area. However, when it is necessary to find the height, h becomes the focus of the formula and thus is isolated using proper algebraic properties. In this lesson, you will rearrange literal equations (equations that involve two or more variables) and formulas with a degree of 2. 2 5.3.4: Rearranging Formulas

3. Key Concepts Literal equations and formulas contain equal signs. Just as in any other equation, we must apply proper algebraic properties to maintain a balance when changing the focus of an equation or formula. That is, if you subtract a value from one side of the equation, you must do the same to the other side, and so on. When you change the focus of a literal equation or formula, you are isolating the variable in question. To isolate a variable that is squared, perform the inverse operation by taking the square root of both sides of the equation. 3 5.3.4: Rearranging Formulas

4. Key Concepts, continued When you take the square root of a real number there are two solutions: one is positive and the other is negative. Take (a) 2, for example. When you square a, the result is a 2. The same, however, is true for (–a) 2. When you square –a, the result is still a 2 because the product of two negatives is a positive. In other words, (–a) 2 = a 2. 4 5.3.4: Rearranging Formulas

5. Key Concepts, continued Since taking a square root involves finding a number that you can multiply by itself to result in the square, we must take into account both the positive and negative. That is, When solving for a squared term of a formula, the focus is important in determining whether to use two solutions. 5 5.3.4: Rearranging Formulas

6. Key Concepts, continued If the focus is any quantity that would never be negative in real life, such as distance, time, or population, it is appropriate to ignore the negative. When solving for a variable in a multi-step equation, first isolate the term containing the variable using subtraction or addition. Then determine which operations are applied to the variable, and undo them in reverse order. 6 5.3.4: Rearranging Formulas

7. Common Errors/Misconceptions forgetting to use the inverse operations in the correct order forgetting that there are likely two solutions (one positive and one negative) when solving for a squared term 7 5.3.4: Rearranging Formulas

8. Guided Practice Example 1 Solve the equation x 2 + y 2 = 100 for y. 8 5.3.4: Rearranging Formulas

9. Guided Practice: Example 1, continued 1.Isolate y. Begin by subtracting x 2 from both sides. 9 5.3.4: Rearranging Formulas x 2 + y 2 = 100Original equation y 2 = 100 – x 2 Subtract x 2 from both sides. Take the square root of both sides. Simplify, remembering that the result could be positive or negative.

10. Guided Practice: Example 1, continued 2.Summarize your result. The formula x 2 + y 2 = 100 can be rewritten as 10 5.3.4: Rearranging Formulas ✔

11. Guided Practice: Example 1, continued 11 5.3.4: Rearranging Formulas

12. Guided Practice Example 2 Solve y = 3(x – 7) 2 + 8 for x. 12 5.3.4: Rearranging Formulas

13. Guided Practice: Example 2, continued 1.Isolate x. 13 5.3.4: Rearranging Formulas y = 3(x – 7) 2 + 8Original equation y – 8 = 3(x – 7) 2 Subtract 8 from both sides. Divide both sides by 3. Simplify.

14. Guided Practice: Example 2, continued 14 5.3.4: Rearranging Formulas Take the square root of both sides. Simplify. Add 7 to both sides.

15. Guided Practice: Example 2, continued 2.Summarize your result. The equation y = 3(x – 7) 2 + 8 solved for x is 15 5.3.4: Rearranging Formulas ✔

16. Guided Practice: Example 2, continued 16 5.3.4: Rearranging Formulas