Simplifying Algebraic Expressions

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Solving Equations

Finding the value of a variable

Solving EquationsAn equation is two expressions separated by an equal sign. Equations work similar to a balance, where the stand is the equal sign and the trays contain the expressions. In

order to keep the trays balanced, you must add, subtract, multiply or divide the same amount to both trays in order to keep it

balanced.

So the next question is “what are we supposed to do mathematically to both sides

of an equation?”

The object of solving equations is to find the VALUE of the variable contained in

the equation.

Keep in mind this goal - Finding the valueof a variable can be accomplished byisolating one, positive, variable on

one side of the equation.

How do you isolate one, positive variable?

In Algebra, we use Properties of Equality to help us to isolate and create one, positive variable.

Properties of Equality allow us to create zero or create one whenever

needed. This is necessary in order to

reach our goal - to isolate one, positive variable.

There are five basic ways that numbers affect variables.

x 121) They can be added to the variable

There are five basic ways that numbers affect variables.

1) They can be added to the variable

2) They can be subtracted from the variable

x 23

There are five basic ways that numbers affect variables.

1) They can be added to the variable

2) They can be subtracted from the variable

3) They can be multiplied by a variable

3x

There are five basic ways that numbers affect variables.

1) They can be added to the variable

2) They can be subtracted from the variable

3) They can be multiplied by a variable

4) The variable can be divided by a number x

5

There are five basic ways that numbers affect variables.

1) They can be added to the variable

2) They can be subtracted from the variable

3) They can be multiplied by a variable

4) The variable can be divided by a number

5) The variable can be multiplied by a fraction

34x

Tools that you will use to aide you in isolating and creating one, positive variable are called

properties of equality. When using these properties they always apply to both sides of the equation. This

is why they work. They always keep an equation balanced.

A number added to a variable can dissapear by adding it’s opposite.

Addition/Subtraction Property of Equality

A number added to a variable can dissapear by adding it’s opposite.

Addition/Subtraction Property of Equality

A number multiplied by a variable can be changed to “1” by dividing the number by itself.

Division Property of Equality

A number added to a variable can dissapear by adding it’s opposite.

Addition/Subtraction Property of Equality

A number multiplied by a variable can be changed to “1” by dividing the number by itself.

Division Property of Equality

A number divided by a variable can be changed to “1” by multiplying the number by itself.

Multiplication Property of Equality

A number added to a variable can dissapear by adding it’s opposite.

Addition/Subtraction Property of Equality

A number multiplied by a variable can be changed to “1” by dividing the number by itself.

Division Property of Equality

A number divided by a variable can be changed to “1” by multiplying the number by itself.

Multiplication Property of Equality

Any fraction times it’s reciprocal is 1.Property of Recipricols

Here are the 5 basic kinds of problems.

Problem type 1 - a number is added to x.

x 12 4

Here are the 5 basic kinds of problems.

Problem type 1

x 12 4 12 12

the opposite of addition issubtraction

Here are the 5 basic kinds of problems.

Problem type 1

x 12 4 12 12

x 8

Here are the 5 basic kinds of problems.

Problem type 2 - a number is subtracted from xx 6 9

Here are the 5 basic kinds of problems.

Problem type 2

x 6 9

6 6

the opposite ofsubtraction is addition

Here are the 5 basic kinds of problems.

Problem type 2

x 6 9

6 6

x 3

Here are the 5 basic kinds of problems.

Problem type 3 - x is multiplied by a number

12x 36

Here are the 5 basic kinds of problems.

Problem type 3 12x 36

12x 12

36 12

the opposite of multiplication is division

Here are the 5 basic kinds of problems.

Problem type 3

12x 36

12x 12

36 12

x 3

Here are the 5 basic kinds of problems.

Problem type 4 - x is divided by a number

x 7

3

Here are the 5 basic kinds of problems.

Problem type 4x 7

3

71

x 7

3 7 the opposite of division is multiplication

Here are the 5 basic kinds of problems.

Problem type 4x

73

71

x 7

3 7

x 21

Here are the 5 basic kinds of problems.

Problem type 5 - a fraction is multiplied by x

23x 12

Here are the 5 basic kinds of problems.

Problem type 5

23x 12

32

23x

121

32

any number times it’s reciprocal is“1”

Here are the 5 basic kinds of problems.

Problem type 5

23x 12

32

23x

121

32

x 2 2 3 3

2

Here are the 5 basic kinds of problems.

Problem type 5

2

3x 12

32

23x

121

32

x 2 2 3 3 2

x 18

Solving Proportionsx8

42

Solving Proportionsx8

42

2x 8 4

Solving Proportionsx8

42

2x 8 4 2x 32

Solving Proportionsx8

42

2x 8 4 2x 32

2x2

32

2

Solving Proportionsx8

42

2x 8 4 2x 32

2x2

322

x 16

Two step problems(two basic problems in one)

Simplify anything that can be simplified. ? Any like terms on the same side of the equal sign?? Any multiplication or distributive property to do?

Get rid of anything that is added or subtracted to the variable.

Create “1” positive variable.

Two step problems(two basic problems in one)

2x 6 12

Two step problems(two basic problems in one)

2x 6 126 6

Two step problems(two basic problems in one)

2x 6 126 6

2x 6

Two step problems(two basic problems in one)

2x 6 126 6

2x 6 2x 2

6 2

Two step problems(two basic problems in one)

2x 6 126 6

2x 6 2x 2

6 2

x 3

Two step problems(two basic problems in one)

35x 8 17

8 8

3

5x 9

53

35x

91

53

x 3 3 5

3x 15

Two step problems(two basic problems in one)

35x 8 17

8 8

3

5x 9

53

35x

91

53

x 3 3 5

3x 15

Two step problems(two basic problems in one)

35x 8 17

8 8

3

5x 9

53

35x

91

53

x 3 3 5

3x 15

Two step problems(two basic problems in one)

35x 8 17

8 8

3

5x 9

53

35x

91

53

x 3 3 5

3x 15

Two step problems(two basic problems in one)

35x 8 17

8 8

3

5x 9

53

35x

91

53

x 3 3 5

3x 15

Two step problems(two basic problems in one)

35x 8 17

8 8

3

5x 9

53

35x

91

53

x 3 3 5

3x 15

Solving two-step Proportionsx 3

4

42

2 x 3 4 4 2x 6 16

6 6

2x 22

2x2

222

x 11

Solving two-step Proportionsx 3

4

42

2 x 3 4 4 2x 6 16

6 6

2x 22

2x2

222

x 11

Solving two-step Proportionsx 3

4

42

2 x 3 4 4 2x 6 16

6 6

2x 22

2x2

222

x 11

Solving two-step Proportionsx 3

4

42

2 x 3 4 4 2x 6 16

6 6

2x 22

2x2

222

x 11

Solving two-step Proportionsx 3

4

42

2 x 3 4 4 2x 6 16

6 6

2x 22

2x2

222

x 11

Solving two-step Proportionsx 3

4

42

2 x 3 4 4 2x 6 16

6 6

2x 22

2x2

222

x 11

Solving two-step Proportionsx 3

4

42

2 x 3 4 4 2x 6 16

6 6

2x 22

2x2

222

x 11

Equations with variables on both sides of the equation.

2x 5 6x 25

6x 6x

4x 5 25

5 5

4x 20

4x4

204

x 5

Equations with variables on both sides of the equation.

2x 5 6x 25

6x 6x

4x 5 25

5 5

4x 20

4x4

204

x 5

Equations with variables on both sides of the equation.

2x 5 6x 25

6x 6x

4x 5 25

5 5

4x 20

4x4

204

x 5

Equations with variables on both sides of the equation.

2x 5 6x 25

6x 6x

4x 5 25

5 5

4x 20

4x4

204

x 5

Equations with variables on both sides of the equation.

2x 5 6x 25

6x 6x

4x 5 25

5 5

4x 20

4x4

204

x 5

Equations with variables on both sides of the equation.

2x 5 6x 25

6x 6x

4x 5 25

5 5

4x 20

4x4

204

x 5

Equations with variables on both sides of the equation.

2x 5 6x 25

6x 6x

4x 5 25

5 5

4x 20

4x4

204

x 5

Java applets to practice more.

•Algebra Balance Scales

•Algebra Balance Scales - Negatives