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Solving Equations
Finding the value of a variable
Solving Equations
An 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 value
of a variable can be accomplished by
isolating 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.
1) They can be added to the variable
x 12
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 3
a fraction
4
x
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 is
subtraction
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 x
x 6  9
Here are the 5 basic kinds of
problems.
Problem type 2
x  6  9
6 6
the opposite of
subtraction 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 36

12 12
the opposite of
multiplication is
division
Here are the 5 basic kinds of
problems.
Problem type 3
12x  36
12x 36

12 12
x 3
Here are the 5 basic kinds of
problems.
Problem type 4 - x is divided by a number
x
3
7
Here are the 5 basic kinds of
problems.
Problem type 4
x
3
7
7 x
   37
 1  7
the opposite of
division is
multiplication
Here are the 5 basic kinds of
problems.
Problem type 4
x
3
7
7 x
   37
 1  7
x  21
Here are the 5 basic kinds of
problems.
Problem type 5 - a fraction is multiplied by x
2
 x  12
3
Here are the 5 basic kinds of
problems.
Problem type 5
2
 x  12
any number
3
times it’s
 3  2  12  3  reciprocal is
  x     “1”
 2  3  1  2 
Here are the 5 basic kinds of
problems.
Problem type 5
2
 x  12
3
 3  2  12  3 
  x   
 2  3  1  2 
22 33
x 
2
Here are the 5 basic kinds of
problems.
Problem type 5
2
 x  12
3
 3  2  12  3 
  x   
 2  3  1  2 

2233
x
2
x  18
Solving Proportions
x 4

8
2
Solving Proportions
x 4

8
2
2x  84
Solving Proportions
x 4

8
2
2x  84
2x  32
Solving Proportions
x 4

8 2
2x  84
2x  32
2x 32

2
2
Solving Proportions
x 4

8
2
2x  84
2x  32
2x 32

2
2
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  12
6 6
Two step problems
(two basic problems in one)
2x 6  12
6 6
2x  6
Two step problems
(two basic problems in one)
2x 6  12
6 6
2x  6
2x 6

2 2
Two step problems
(two basic problems in one)
2x 6  12
6 6
2x  6
2x 6

2 2
x 3
Two step problems
(two basic problems in one)
3
 x  8  17
5
8 8
3
 x 9
5
 5 3  9  5 
  x   
 3 5  1  3 
x
335
x  15
3
Two step problems
(two basic problems in one)
3
 x  8  17
5
8 8
3
 x 9
5
 5 3  9  5 
  x   
 3 5  1  3 
x
335
x  15
3
Two step problems
(two basic problems in one)
3
 x  8  17
5
8 8
3
 x 9
5
 5 3  9  5 
  x   
 3 5  1  3 
x
335
x  15
3
Two step problems
(two basic problems in one)
3
 x  8  17
5
8 8
3
 x 9
5
 5 3  9  5 
  x   
 3 5  1  3 
x
335
x  15
3
Two step problems
(two basic problems in one)
3
 x  8  17
5
8 8
3
 x 9
5
 5 3  9  5 
  x   
 3 5  1  3 
x
335
x  15
3
Two step problems
(two basic problems in one)
3
 x  8  17
5
8 8
3
 x 9
5
 5 3  9  5 
  x   
 3 5  1  3 
x
335
x  15
3
Solving two-step Proportions
x 3 4

4
2
2 x  3  4 4
2x  6  16
6 6
2x  22
2x 22

2
2
x  11
Solving two-step Proportions
x 3 4

4
2
2 x  3  4 4
2x  6  16
6 6
2x  22
2x 22

2
2
x  11
Solving two-step Proportions
x 3 4

4
2
2 x  3  4 4
2x  6  16
6 6
2x  22
2x 22

2
2
x  11
Solving two-step Proportions
x 3 4

4
2
2 x  3  4 4
2x  6  16
6 6
2x  22
2x 22

2
2
x  11
Solving two-step Proportions
x 3 4

4
2
2 x  3  4 4
2x  6  16
6 6
2x  22
2x 22

2
2
x  11
Solving two-step Proportions
x 3 4

4
2
2 x  3  4 4
2x  6  16
6 6
2x  22
2x 22

2
2
x  11
Solving two-step Proportions
x 3 4

4
2
2 x  3  4 4
2x  6  16
6 6
2x  22
2x 22

2
2
x  11
Equations with variables on
both sides of the equation.
2x  5  6x  25
6x
6x
4x  5  25
5
5
4x  20
4x 20

4
4
x  5
Equations with variables on
both sides of the equation.
2x  5  6x  25
6x
6x
4x  5  25
5
5
4x  20
4x 20

4
4
x  5
Equations with variables on
both sides of the equation.
2x  5  6x  25
6x
6x
4x  5  25
5
5
4x  20
4x 20

4
4
x  5
Equations with variables on
both sides of the equation.
2x  5  6x  25
6x
6x
4x  5  25
5
5
4x  20
4x 20

4
4
x  5
Equations with variables on
both sides of the equation.
2x  5  6x  25
6x
6x
4x  5  25
5
5
4x  20
4x 20

4
4
x  5
Equations with variables on
both sides of the equation.
2x  5  6x  25
6x
6x
4x  5  25
5
5
4x  20
4x 20

4
4
x  5
Equations with variables on
both sides of the equation.
2x  5  6x  25
6x
6x
4x  5  25
5
5
4x  20
4x 20

4
4
x  5
Java applets to practice more.
•Algebra Balance Scales
•Algebra Balance Scales - Negatives