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Equations How to Solve Them

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7/8/2013 Equations 2 What Are Equations ? Equation Statement that two expressions are equal What do we DO with expressions ? Simplify them Evaluate them What do we DO with equations ? Solve them

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7/8/2013 Equations 3 What Are Equations ? Three equation categories Identity: Logically True Example: 2(x + 1) = 2x + 2 True for ALL values of x Logically False Example: x + 3 = x NOT true for ANY x ( would imply 3 = 0, a contradiction ! )

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7/8/2013 Equations 4 What Are Equations ? The third equation category Conditional Equations Example: 3x + 1 = 2x – 7 True for SOME values of x (x = – 6 in this case) False for other values of x

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7/8/2013 Equations 5 Linear Equations in 1 Variable Standard Form: for some constants a, b with a ≠ 0 Solutions Solution is a value of x that makes the equation TRUE A solution is a number WHY a ≠ 0 ? a x + b = 0

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7/8/2013 Equations 6 Linear Equations in 1 Variable Equivalent Equations Equations are equivalent if and only if they have the same solutions Solving an equation transforms it into an equivalent equation of form: The number r is the solution x = r WHY ? – there is only one

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7/8/2013 Equations 7 Cancellation Rule for Addition a + c = b + c if and only if a = b for any numbers a, b and c Cancellation Rule for Multiplication ac = bc if and only if a = b for any numbers a, b and c with c ≠ 0 Techniques Solving Equations

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7/8/2013 Equations 8 Examples Example 1. 2x + 3 = 7 Example 2. = 2 2 Techniques for Solving Examples = then 2x = 4 then x = 2 If 2x = 4 Cancellation Rule for Addition Cancellation Rule for Multiplication

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7/8/2013 Equations 9 Addition Rule a = b if and only if a + c = b + c for any numbers a, b and c Multiplication Rule a = b if and only if ac = bc for any numbers a, b and c with c ≠ 0 Same Rules in Reverse Question: Why can we add 0 but not multiply by 0 ?

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7/8/2013 Equations 10 Example 1. If 2x – 3 = 7 Example 2. If 2x = 10 Same Rules in Reverse then 2x – = so 2x = 10 so x = 5 Question: Why can we add 0 but not multiply by 0 ? then (2x) = (10) Addition Rule Multiplication Rule

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7/8/2013 Equations 11 Solve: –3(2x – 1) = 2x – 6x + 3 = 2x distributive property – 6x x = 2x + 6x addition rule 3 = 8x simplification (1/8)(3) = (1/8)(8x) multiplication rule 3/8 = x simplification Solution is Solving Symbolically 3 8 The solution is NOT WHY ? Solution Set: { } 3 8 Note: 3 8 x =

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7/8/2013 Equations 12 Solve: Simplify by clearing fractions Solving Symbolically – 2 3x – 1 5 = 2 – x 3 ( ) – 2 3x – = ( ) 2 – x (3x – 1) – 30 = 5(2 – x) = ( ) – x ( ) 3x – – ( )

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7/8/2013 Equations 13 Solve: Solving Symbolically – 2 3x – 1 5 = 2 – x 3 3(3x – 1) – 30 = 5(2 – x) 9x – 33 = 10 – 5x 14x = 43 Equivalent Equation Solution : = x Solution set : { }

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7/8/2013 Equations 14 Solve: 3x + 1 = –2x + 11 Consider this as the equality of two functions y 1 and y 2 with and Lines intersect where (x, y 1 ) = (x, y 2 ) Solving Equations Graphically y x y 1 = 3x + 1 y 2 = –2x + 11 y1y1 y2y2 x

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7/8/2013 Equations 15 Solve: 3x + 1 = –2x + 11 Lines intersect where (x, y 1 ) = (x, y 2 ) In this case (x, y 1 ) = (x, y 2 ) = (2, 7) Solving Equations Graphically y x y 1 = 3x + 1 y 2 = –2x + 11 y1y1 y2y2 x Sox = 2 Solution is 2 Solution set is { 2 } Question: How do we find 2 and 7 graphically ?

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7/8/2013 Equations 16 Solve: 14x – 36 = 7 This can be written as y(x) = 14x – 36 = 7 Want x value where y = 7 Desired x between 3 and 4 Increase resolution between 3 and 4 Solving Equations Numerically 0 –36 1 –22 2 – x y 7

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7/8/2013 Equations 17 Solve: 14x – 36 = 7 Increased resolution Solving Equations Numerically x y y = 7 for x between 3.0 and 3.1 Continue refining x Expand 3.0 – 3.1 into new table (3.00 – 3.09) for next decimal on y *

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7/8/2013 Equations 18 * Solve: 14x – 36 = 7 Solving Equations Numerically x y …. … Continue refining x Question: How accurate is this method ? How long does it take ? to force y closer to 7

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7/8/2013 Equations 19 Graphical Solution Least accurate, visual solution Can be automated via computer/calculator Makes trends more obvious Numerical Solution Approximate solution but refinable Natural for collected data Easily automated Solving Equations: Review

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7/8/2013 Equations 20 Symbolic Solution The most accurate Purely algebraic Good for predictions Solving Equations: Review

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7/8/2013 Equations 21 Think about it !

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