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IntroductionThe following pages and links combine to form the next unit. This unit is called Solving One-Variable Equations. The lesson goes in conjunction with Chapter 3 in your Algebra 1 book.

This lesson is extremely important in understanding math from now until you graduate from high school and eventually college, so please take this seriously.

Symbols and VariablesBefore getting into actually solving the equations, we will first discuss the importance of symbols and variables.

SYMBOL - something that is used to represent something else

Go to the next slide and take note of the symbols!

Examples of Symbols Note what the following pictures represent:

America ReligionPirates

More Symbols Money STOP The Number 4

Whats a Variable?a VARIABLE is just a letter that represents a number!

For instance, x = 4 makes x a symbol that represents 4 in this example!

All letters can be used as a symbol in algebra.

Laws of MultiplicationIdentity PropertySays that 1 a = aCommutative PropertySays that a b = b a Distributive PropertySays that a(b + c) = ab + acAssociative Propertya (b c) = (a b) c

Other Key VocabularyCoefficientThe number multiplied by the variable in a termExamples

The Coefficient Here is 4.

The Coefficient Here is 3.4x3x 1

Other Key Vocabulary 2ConstantThe number added to or subtracted to the variable in a termThis number has no variable attached to it (like a coefficient does).Examples

The Constant Here is -1.

The Constant Here is 7.6x + 73x 1

Other Key Vocabulary 3EquivalentEquations with the same solutions as the original one.

Example:

x + 7 and y 3 both equal 5 in this case, making the expressions (x + 7) and (y 3)equivalent.

x + 7 = 5y 3 = 5

Other Key Vocabulary 4ReciprocalYou will get 1 if you multiply a number by its reciprocal (or all of the terms will cancel out.Ex. a/b has the reciprocal b/aEx2:

Are reciprocals of one another because:3/2 times 2/3 = 1And because one is the inverse of the other3/22/3

Question 0.1:Solve the following EquationBy the Associative Property, x (y z) equals which of the followingx (y z)

(x y z)

(x y) z

((x ((y) z)))

Solving UsingAddition &Subtraction

Addition and Subtraction 1.2Solve x 5 = -13x 5 = -13Rewrite the original equation

x 5 + 5 = -13 + 5Add 5 to each side.

x = -8.Simplify, and our solution is x = -8.

Addition and Subtraction 1.3 Now Check Your Answer x 5 = -13Rewrite the original equation

-8 - 5 = -13Substitute -8 in for x.

-13 = -13The Solution is Correct!

SimplifyDid you understand what was meant by the word simplify?

Definition: SimplifyTo Take an equation down to the simplest terms.Ex: we had x 5 + 5 = -13 + 5 in the earlier example.Obviously -5 + 5 = 0, so those terms essentially CANCEL.Additionally, -13 + 5 = -8Therefore you could write x 5 + 5 = -13 + 5 more simply as x = -8.

Addition and Subtraction 1.4Now Try Another Example! Solve r + 3 = 2r + 3 = 2 Rewrite the original equation

r + 3 3 = 2 3 Subtract 3 from each side.

r = -1.Simplify, and our solution is r = -1.

Addition and Subtraction 1.5 Now Check Your Answerr + 3 = 2Rewrite the original equation -1 + 3 = 2Substitute -1 in for r.

2 = 2The Solution is Correct!

Multiplication and Division 2.2Solve -4x = 16-4x = 16Rewrite the original equation

-4x = 16-4 -4Divide each side by -4

x = -4.Simplify, and our solution is x = -4.

Multiplication and Division 2.3 Now Check Your Answer -4x = 16Rewrite the original equation

-4 (-4) = 16Substitute -4 in for x.

-16 = -16The Solution is Correct!

Multiplication and Division 2.4Try Another! Solve x = 5 x = 5 Rewrite the original equation

x (2) = 5 (2) Multiply each side by 2.

x = 10.Simplify, and our solution is x = 10.

Multiplication and Division 2.5 Now Check Your Answerx = 5 Rewrite the original equation (10) = 5 Substitute 10 in for x.

5 = 5The Solution is Correct!

Question 2.1:Solve the following Equation3x= 12x = 36

x = 9

x = 15

x = 4

Multi-Step Equations 3.1Multi-Step Equations are equations that include solving by addition and/or subtraction as well as by multiplication and/or division.Once again, we will proceed by example.Try the following: Solve 7x + 6 = -8

Multi-Step Equations 3.2We will start manipulating the equation by doing the addition and subtraction part.7x + 6 = -8Rewrite the original equation

7x + 6 6 = -8 6 Subtract 6 from each side.

7x = -14Simplify.

Multi-Step Equations 3.3Now well do the multiplication part. Remember, from the last slide we were left with 7x = -14

7x = -14 7 7Divide each side by 7

x = -2.Simplify once more, and our solution is x = -2.

Multi-Step Equations 3.4Now Check Your Answer7x +6 = -8Rewrite the original equation

7(-2) +6 = -8Substitute -2 in for x.

-14 +6 = -8Distribute (multiply the 7 and -2).

-8 = -8The Solution is Correct!

Multi-Step Equations 3.5Now Try Another ExampleSolve 7x 3x 8 = 24 7x 3x 8 = 24Rewrite the original equationNotice that there are 2 x terms.

4x 8 = 24Combine Like Terms

4x 8 + 8 = 24 + 8Add 8 to each side.

Multi-Step Equations 3.64x = 32Simplify.

4x = 324 4Divide each side by 4.x = 8.Simplify once more to get x = 8.

Multi-Step Equations 3.7Now Check Your Answer.7x 3x 8 = 24Rewrite the original equation

7(8) 3(8) 8 = 24Substitute 8 in for EACH x.

56 24 8 = 24Distribute.

24 = 24The Solution is Correct!

Question 2.1:Solve the following Equation5 4x = 8 x x = 5

x = -1

x = 3

x = 8

Word Problems 4.1A word problem is just a paragraph version of an equation.

The tricky part is to determine what the word problem means in equation terms.

Once you figure out how to do that part, it becomes either an addition/subtraction, multiplication/division problem or a multi-step problem.

Word Problems 4.2In order to solve word problems, the best way, again, is to proceed by example.

Try this one:Jason sells chocolate bars for his baseball team. Each chocolate bar costs 2 dollars. If at the end of the day he had 38 dollars, how many chocolate bars did Jason sell?

Word Problems 4.3Word ProblemHow to solveJason sells chocolate bars for his baseball team. Each chocolate bar costs 2 dollars. If at the end of the day he had 38 dollars, how many chocolate bars did Jason sell?

First, identify all of the important facts/numbers in the problem.We see that Jason sold each chocolate bar for 2 dollars and ended up with 38 dollars

Word Problems 4.4Word ProblemHow To SolveJason sells chocolate bars for his baseball team. Each chocolate bar costs 2 dollars. If at the end of the day he had 38 dollars, how many chocolate bars did Jason sell?

Now that we have these facts, we can derive the equationSince each bar costs 2 dollars, then we know 2 will be in the problem somehow

Word Problems 4.5Word ProblemHow To SolveJason sells chocolate bars for his baseball team. Each chocolate bar costs 2 dollars. If at the end of the day he had 38 dollars, how many chocolate bars did Jason sell?

We want to MULTIPLY the 2 by our variable (call it x).Therefore, a 2x will appear somewhere in the equation that we want to solve

Word Problems 4.6Word ProblemHow To solveJason sells chocolate bars for his baseball team. Each chocolate bar costs 2 dollars. If at the end of the day he had 38 dollars, how many chocolate bars did Jason sell?

Finally, 38 is the result of our equation.Combing the two together, we can derive the following equation:2x = 38

Word Problems 4.7Now that we have our equation, we can solve it just like we did earlier.2x = 38 Rewrite the original equation

2x = 38 2 2Divide each side by 2

x = 19.Simplify, and our solution is x = 19.

Word Problems 4.8Now Check Your Answer 2x = 38Rewrite the original equation

2 (19) = 38Substitute 19 in for x.

38 = 38The Solution is Correct!

Word Problems 4.9The last component of a word problem is a summary sentenceDefinition:Summary SentenceA sentence that goes with word problems that sums up your answer in sentence or word form.

So for our example, x = 19 would be answered as follows:Jason sold a total of 19 chocolate bars.

Question 4.1Now You Try:Kelly has 15 DVDs in her case. After buying more DVDs at The Exchange, she now has 23. How many more CDs did Kelly buy?

Kelly Bought 4 CDsKelly Bought 23 CDs

Kelly Bought 15 CDsKelly Bought 8 CDs

Word Problems 4.11Word ProblemHow To SolveDan and Amber are adding up their ages. Dan is 2 times Ambers age then minus 5. Amber is 9 years old. How old is Dan?

First, identify all of the important facts/numbers in the problem.We see that Amber is 9 years old.We see that Dan is 2 times Ambers age then minus 5.

Word Problems 4.12Word ProblemHow To SolveDan and Amber are adding up their ages. Dan is 2 times Ambers age then minus 5. Amber is 9 years old. How old is Dan?

Since Dan is 2 times Ambers age then minus 5, and Amber is 9, our equation is then2x 5 = 9

Word Problems 4.13Now solve the equation we just got.2x 5 = 9 Rewrite the original equation

2x 5 + 5 = 9 + 5 Add 5 to each side

2x = 14 2 2Divide each side by 2

x = 7.Simplify, and our solution is x = 7.

Word Problems 4.14Now Check Your Answer 2x 5 = 9Rewrite the original equation

2 (7) 5 = 9Substitute 7 in for x.

14 5 = 9Distribute (multiply the 7 and 2).

9 = 9The Solution is Correct!

Word Problems 4.15Dont forget! We have to write a summary sentence for this problem!

Our Summary Sentence is:Therefore, Amber is 9 years old and Dan is 7 years old.

Question 4.2Now Try This problem:Mike sells cakes at a local bakery. He charges 10 dollars for each cake. At the end of the day, there was 180 dollars worth of cake sold. How many cakes did Mike sell?Mike Sold 18 CakesMike Sold 10 Cakes

Mike Sold 170 CakesMike Sold 8 Cakes

Fractions & Decimals 5.1Solving Equations with Fractions and Decimals in them is just the same as solving using addition, subtraction, multiplication, division or a combination, except instead of integers, fractions and decimals are used.

Fractions and Decimals 5.2What is a fraction?A fraction is a number that represents a part of a whole. These numbers are written with a forward slash or horizontal line. Ex: , 2/3, 5/8, NUMERATOR (the number on top)DENOMENATOR (the number on bottom)

Fractions and Decimals 5.3What is a decimal?A decimal is also a number that represents part of a whole number. This time, it is represented with by a number after a period. Ex:1.4, 5.9, .08, .01763, etcWhat is an integer?An integer is a whole number. It is every natural number, (0, 1, 2, 3, ...) and every negative natural number (-1, -2, -3, ...)

Fractions and Decimals 5.4Since you will solve these equations in the same way that you solved the other ones, we will proceed by example.So, lets try this example:Solve 0.2x + 5 = 15

Fractions and Decimals 5.50.2x + 5 = 15 Rewrite the original equation

0.2x + 5 5 = 15 5 Subtract 5 from each side.

0.2x = 10Simplify.

Keep GOING!!!

Fractions and Decimals 5.60.2x = 100.2 0.2Divide each side by 0.2You Can Use Your Calculator if you need it!

x = 50.Simplify once more, and our solution is x = 50.

Question 5.1:Solve the following Equation-1.2x - 3 = 9 x = 1

x = -1

x = 10

x = -10

Fractions and Decimals 5.8x = 1 Rewrite the equation again

x (4) = 1 (4) Multiply each side by 4.Note that 4 is the RECIPROCAL of .And remember two numbers that are reciprocals of one another when multiplied together leaves only 1, thus x is left here.

x = 4.Simplify, and our solution is x = 4.

Question 5.2:Solve the following Equationx + 6 = 9 x = 1

x = 3

x = 6

x = 9