Introduction to Solving Problems Introduction to Solving Problems AlgebraicallyAlgebraically

ObjectivesObjectives 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 describe the difference between “guess and check” and “algebraic solution”“algebraic solution”

• describe, with prompts, the general steps used to solve a describe, with prompts, the general steps used to solve a problem algebraicallyproblem algebraically

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.

An Algebraic SolutionAn Algebraic Solution

Algebra uses letters called “variables” to take the Algebra uses letters called “variables” to take the place of an unknown numberplace of an unknown number

An algebraic solution often includes:An algebraic solution often includes: selection of variablesselection of variables writing algebraic expressions and equationswriting algebraic expressions and equations substitutionsubstitution simplificationsimplification additive inverseadditive inverse multiplicative inversemultiplicative inverse

Where Do I Start?Where Do I Start?

Select variablesSelect variables to represent the unknown numbers to represent the unknown numbers Let m = points scored by MaryLet m = points scored by Mary Let c = points scored by CharlesLet c = points scored by Charles Let a = points scored by AdamLet a = points scored by Adam Let p = points scored by PaulLet p = points scored by Paul

Next Step?Next Step?

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

And Then?And Then?

Substitute.Substitute. Find a value for ‘p’. Find a value for ‘p’. c = 2m – 16,c = 2m – 16, and and a = m + 39a = m + 39 so in our 3 so in our 3rdrd equation we can equation we can

substitute m + 39 for ‘a’ and 2m – 16 for ‘c’substitute m + 39 for ‘a’ and 2m – 16 for ‘c’ a + p = m + c + 18a + p = m + c + 18

(m + 39) + p = m + (2m – 16) + 18(m + 39) + p = m + (2m – 16) + 18

Whew! Now what?Whew! Now what?

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

p = 2m – 37p = 2m – 37

That was simple. Where to?That was simple. Where to?

Review the information.Review the information. c = 2m – 16c = 2m – 16 Charles’ points.Charles’ points. a = m + 39a = m + 39 Adam’s points.Adam’s points. p = 2m – 37p = 2m – 37 Paul’s points.Paul’s points. a + p = m + c + 18a + p = m + c + 18 Used to find Adam’s points.Used to find Adam’s points. m + c + a + p = 658m + c + a + p = 658 All points add to 658.All points add to 658.

• Look! Mary’s points are in all 5 equationsLook! Mary’s points are in all 5 equations

Start over? No kidding!Start over? No kidding!

SubstituteSubstitute equivalents into the last equation. equivalents into the last equation.

m + c + a + p = 658m + c + a + p = 658

m + (2m – 16) + (m + 39) + (2m – 37) = 658m + (2m – 16) + (m + 39) + (2m – 37) = 658

And the second step is …? And the second step is …?

Simplify Simplify by combining like terms. by combining like terms.

mm + + 2m2m – 16– 16 + + mm + 39+ 39 + + 2m2m – 37– 37 = 658 = 658

6m6m –– 1414 = = 658658

Third step again already. Third step again already.

Use theUse the additive inverse additive inverse to simplify across the to simplify across the equal sign.equal sign.

6m – 14 6m – 14 + 14+ 14 = 658 = 658 + 14+ 14

6m = 6726m = 672

Fourth step, and then some. Fourth step, and then some.

Use theUse the multiplicative inverse multiplicative inverse to find the value of ‘m’.to find the value of ‘m’.

●● 6m = 672 6m = 672 ●●

m = 112m = 112

What’s this? A numerical value for a variable? That does it!!What’s this? A numerical value for a variable? That does it!!

1

6

1

6

Back to the EquationsBack 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 = 112m = 112 c = 2m – 16 becomes c = 2(112) – 16, or c = 224 – 16, or c = 208c = 2m – 16 becomes c = 2(112) – 16, or c = 224 – 16, or c = 208 a = m + 39 becomes a = 112 + 39, or a = 151a = m + 39 becomes a = 112 + 39, or a = 151 p = 2m – 37 becomes p = 2(112) – 37, or p = 224 – 37 or p = 187p = 2m – 37 becomes p = 2(112) – 37, or p = 224 – 37 or p = 187

And Finally…And Finally…

Check Check the solution using the final equation.the solution using the final equation.

m + c + a + p = 658m + c + a + p = 658

Mary’s points Mary’s points 112112 Charles’ pointsCharles’ points 208208 Adam’s pointsAdam’s points 151151 Paul’s pointsPaul’s points 187187

658 658

+

So What??So What??(Conclusion)(Conclusion)

So now you’ve solved an algebraic problem.So now you’ve solved an algebraic problem. Key conceptsKey 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.