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Goal: Using Transformations to Solve Linear Equations Agenda Tuesday/Wednesday – Learn how to solve equations with addition and subtraction (3.1). Thursday/Friday – Learn how to solve equations with multiplication and division (3.2). Next Monday – Using both to solve multi-step equations (3.3). What’s a Linear Equation…in one variable? From the glossary – “An equation in which the variable is raised to the first power and does not occur in a denominator, inside a square root symbol, or inside an absolute value symbol.” Essentially their graphs have straight lines. Some Examples: x 1 4 3 1 q 4 2 y 2 6 y 3 y 1 Some non-examples: x4 5 x 1 4 2 y 2 6 x 2 x 1 4 Solving Equations… Transformations You will now solve equations using “transformations”, which are a combination of inverse operations used to isolate the variable to one side of the equation. Transformations may be one-step or multiplestep solutions. Equations may also require simplification to minimize the number of transformations. Solving Equations… Transformations An equation is transformed when it is rewritten to produce an equation with the same solutions as the original equation. Such equations are said to be “equivalent”. Examples: x42 and x 2 and x20 Solving Equations… Transformations using Addition and Subtraction Examples: Examples: For 10.28.10 HW Do Problems: Regular: pg. 135ff…4-12 (even);21-29 (odd); 42-46 (even); Fraction Practice – 2-8 (even); 17,20,26,27; Advanced: pg.135ff…28-40(even); 49,52,53,55; Fraction Practice – 5-8 (all); 18,21,27,30; 11.2.10 Warm-Up 11.2.10 - We’ve already seen how to solve linear equations using transformations of addition and subtraction…that is, inverse operations and simplification. Now we will… Solve Linear Equations w/division and multiplication (3.2). Solve linear equations with multi-step transformations (3.3). Solve linear equations with variables on both sides of equations (3.4). Solving Equations… Transformations using Multiplication and Division Examples – Div./Multi. 4n 24 1 y 6 5 1 2 y5 3 3 Do Problems: Pg. 142…39-43 (odd); 49-53 (odd); 58-59 Solving Multiple-Step Equations…Warm-Up Solving Multiple-Step Equations… Solving Linear Equations may require two or more transformations…here’s a simple process to remember: 1.) Collect variables on “left side”; Or collect variables on “right side” of equation. 2.) Simplify one or both sides of the equation (if needed). 3.) Use inverse operations to isolate the variable. Solving Multiple-Step Equations… Examples – Multi-Step: 1 30 16 x 5 7 x 4 x 9 4 (2 x 4) 48 9 Do Problems: Reg. Algebra: Pg. 148-9…25-37 (odd); 5054 (even) Adv. Algebra: pg. 148-9…(33-39 odd); 6265 (all) Solving Linear Equations: Variables on Both Sides Solving Linear Equations: Variables on Both Sides Examples: Special Cases: Identities and No Solutions Collect Variables on Left Side or Collect Variables on Right Side. Odd cases: Identities and cases where there is no solution. Examples: An Identity: 3(4 4 x) 12 x 12 No Solution: 12c 4 12c Do Problems: 11.8.10 HW Reg. Algebra - Pg. 157…1,4-8 (even); 12-24 (even); 45,46,70 OR Pg. 157…29-41 (odd), 47-49 (all); Formulas and Functions The ability to transform and solve equations allows us to use formulas for discovering new information from real-life situations. “A formula is an algebraic equation which relates two or more real-life quantities.” “An equation is in function form if one variable is isolated on one side of the equation.” Formulas and Functions Examples: Al w 5 C ( F 32) 9 D R T I P r t Acircle r 2 Examples: Do Problems: 11.8.10 HW Adv. Algebra – pg. 177…3-21 (odd); 37-42 (all) OR Pg. 177…11-29 (odd), 40-44 (all), 47-49 (all)