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Introduction to Solving Problems Algebraically Objectives Objectives At the end of this lesson, you will be able to: At the end of this lesson, you will be able to: describe the difference between guess and check and algebraic solutiondescribe the difference between guess and check and algebraic solution describe, with prompts, the general steps used to solve a problem algebraicallydescribe, with prompts, the general steps used to solve a problem algebraically

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Take five minutes to guess the solution to this problem. Your solution should include the number of points scored by each of the four named players. Here is a problem for you to solve. It is not an easy one.

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An Algebraic Solution Algebra uses letters called variables to take the place of an unknown number Algebra uses letters called variables to take the place of an unknown number An algebraic solution often includes: An algebraic solution often includes: selection of variables selection of variables writing algebraic expressions and equations writing algebraic expressions and equations substitution substitution simplification simplification additive inverse additive inverse multiplicative inverse multiplicative inverse

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Where Do I Start? Select variables to represent the unknown numbers Select variables to represent the unknown numbers Let m = points scored by Mary Let m = points scored by Mary Let c = points scored by Charles Let c = points scored by Charles Let a = points scored by Adam Let a = points scored by Adam Let p = points scored by Paul Let p = points scored by Paul

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Next Step? Write algebraic expressions and/or equations Write algebraic expressions and/or equations c = 2m – 16 c = 2m – 16 a = m + 39 a = m + 39 a + p = m + c + 18 a + p = m + c + 18 m + c + a + p = 658 m + c + a + p = 658

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And Then? Substitute. Find a value for p. Substitute. Find a value for p. c = 2m – 16, and a = m + 39 so in our 3 rd equation we can substitute m + 39 for a and 2m – 16 for c c = 2m – 16, and a = m + 39 so in our 3 rd equation we can substitute m + 39 for a and 2m – 16 for c a + p = m + c + 18 a + p = m + c + 18 (m + 39) + p = m + (2m – 16) + 18 (m + 39) + p = m + (2m – 16) + 18

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Whew! Now what? Simplify. Combine like terms on both sides of the equal sign using the additive inverse. Simplify. Combine like terms on both sides of the equal sign using the additive inverse. m + 39 + p = m + 2m – 16 + 18 m + 39 + p = m + 2m – 16 + 18 m – m + 39 – 39 + p = m – m + 2m – 16 + 18 – 39 m – m + 39 – 39 + p = m – m + 2m – 16 + 18 – 39 p = 2m – 37 p = 2m – 37

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That was simple. Where to? Review the information. Review the information. c = 2m – 16Charles points. c = 2m – 16Charles points. a = m + 39Adams points. a = m + 39Adams points. p = 2m – 37Pauls points. p = 2m – 37Pauls points. a + p = m + c + 18Used to find Adams points. a + p = m + c + 18Used to find Adams points. m + c + a + p = 658All points add to 658. m + c + a + p = 658All points add to 658. Look! Marys points are in all 5 equations Look! Marys points are in all 5 equations

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Start over? No kidding! Substitute equivalents into the last equation. Substitute equivalents into the last equation. m + c + a + p = 658 m + (2m – 16) + (m + 39) + (2m – 37) = 658 m + (2m – 16) + (m + 39) + (2m – 37) = 658

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And the second step is …? Simplify by combining like terms. Simplify by combining like terms. m + 2m – 16 + m + 39 + 2m – 37 = 658 m + 2m – 16 + m + 39 + 2m – 37 = 658 6m – 14 = 658 6m – 14 = 658

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Third step again already. Use the additive inverse to simplify across the equal sign. Use the additive inverse to simplify across the equal sign. 6m – 14 + 14 = 658 + 14 6m – 14 + 14 = 658 + 14 6m = 672 6m = 672

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Fourth step, and then some. Use the multiplicative inverse to find the value of m. Use the multiplicative inverse to find the value of m. 6m = 672 6m = 672 m = 112 m = 112 Whats this? A numerical value for a variable? That does it!! 1616 1616

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Back to the Equations Substitute the value of m, 112, for m wherever you see it. Substitute the value of m, 112, for m wherever you see it. m = 112 m = 112 c = 2m – 16 becomes c = 2(112) – 16, or c = 224 – 16, or c = 208 c = 2m – 16 becomes c = 2(112) – 16, or c = 224 – 16, or c = 208 a = m + 39 becomes a = 112 + 39, or a = 151 a = m + 39 becomes a = 112 + 39, or a = 151 p = 2m – 37 becomes p = 2(112) – 37, or p = 224 – 37 or p = 187 p = 2m – 37 becomes p = 2(112) – 37, or p = 224 – 37 or p = 187

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And Finally… Check the solution using the final equation. Check the solution using the final equation. m + c + a + p = 658 m + c + a + p = 658 Marys points 112 Marys points 112 Charles points 208 Charles points 208 Adams points 151 Adams points 151 Pauls points 187 Pauls points 187 658 658 +

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So What?? (Conclusion) So now youve solved an algebraic problem. So now youve solved an algebraic problem. Key concepts Key concepts Choosing variablesChoosing variables Writing algebraic expressions and equationsWriting algebraic expressions and equations SubstitutionSubstitution SimplificationSimplification Additive inverseAdditive inverse Multiplicative inverseMultiplicative inverse Over the next few weeks, you will learn to Over the next few weeks, you will learn to solve algebraic problems by yourself. solve algebraic problems by yourself.

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SOLVING SYSTEMS ALGEBRAICALLY SECTION 3-2. SOLVING BY SUBSTITUTION 1) 3x + 4y = 12 STEP 1 : SOLVE ONE EQUATION FOR ONE OF THE VARIABLES 2) 2x + y = 10.

SOLVING SYSTEMS ALGEBRAICALLY SECTION 3-2. SOLVING BY SUBSTITUTION 1) 3x + 4y = 12 STEP 1 : SOLVE ONE EQUATION FOR ONE OF THE VARIABLES 2) 2x + y = 10.

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