Transcript Block 22M - Math GR. 6-8

Identity and Inverse Properties
Identity Property of
Addition
The Identity Property of Addition states that
for any number x, x + 0 = x
5+0=5
27 + 0 = 27
4.68 + 0 =
¾+0=¾
Identity Property of Multiplication
The Identity Property of Multiplication states
that for any number x, x (1) = x
Remember the number 1 can be in ANY
form.
The number 1 can be in ANY form. In
this case 3/3 is the same as 1.
2 3 6  2 


33 9  3
same
Inverse Property of Addition
The inverse property of addition states that
for every number x, x + (-x) = 0
4 and -4 are considered opposites.
4 + -4 = 0
-4
+4
What number can be added to 15 so
that the result will be zero?
-15
What number can be added to -22
so that the result will be zero?
22
Inverse Property of Multiplication
The Inverse Property of Multiplication states
for every non-zero number n, n (1/n) = 1
The non-zero part is important or else we
would be dividing by zero and we CANNOT
do that.
Properties of Equality
In all of the following properties
Let a, b, and c be real numbers
Properties of Equality
 Addition property:
If a = b, then a + c = b + c
 Subtraction property:
If a = b, then a - c = b – c
 Multiplication property:
If a = b, then ca = cb
 Division property:
a
b
If a = b, then  for c ≠ 0
c c
Addition Property
This is the property that allows you to add the same number
to both sides of an equation.
STATEMENT
REASON
x=y
given
x+3=y+3
Addition property of
equality
Subtraction Property
This is the property that allows you to subtract the same
number to both sides of an equation.
STATEMENT
REASON
a=b
given
a-2=b-2
Subtraction property of
equality
Multiplication Property
This is the property that allows you to multiply the same
number to both sides of an equation.
STATEMENT
REASON
x=y
given
3x = 3y
Multiplication property of
equality
Division Property
This is the property that allows you to divide the same
number to both sides of an equation.
STATEMENT
REASON
x=y
given
x/3 = y/3
Division property of equality
More Properties of Equality
 Reflexive Property:
a=a
 Symmetric Property:
If a = b, then b = a
 Transitive Property:
If a = b, and b = c, then a = c
Substitution Property of
Equality
If a = b, then a may be substituted for b in any equation
or expression.
You have used this many times in algebra.
STATEMENT
x=5
3+x=y
3+5=y
REASON
given
given
substitution
property of equality
Solving One-Step
Equations
Definitions
Term: a number, variable or the
product or quotient of a number
and a variable.
examples:
12
z
2w
c
6
Terms are separated by addition (+)
or subtraction (-) signs.
3a – ¾b + 7x – 4z + 52
How many Terms do you see?
5
Definitions
Constant: a term that is a number.
Coefficient: the number value in
front of a variable in a term.
3x – 6y + 18 = 0
What are the coefficients? 3 , -6
What is the constant?
18
Solving One-Step Equations
A one-step equation means you only have to
perform 1 mathematical operation to solve it.
You can add, subtract, multiply or divide to
solve a one-step equation.
The object is to have the variable by itself on
one side of the equation.
Example 1: Solving an addition equation
t + 7 = 21
To eliminate the 7 add its opposite to both sides of the
equation.
t + 7 = 21
t + 7 -7 = 21 - 7
t + 0 = 21 - 7
t = 14
Example 2:
Solving a subtraction equation
x – 6 = 40
To eliminate the 6 add its opposite to both
sides of the equation.
x – 6 = 40
x – 6 + 6 = 40 + 6
x = 46
Example 3:
Solving a multiplication equation
8n = 32
To eliminate the 8 divide both sides of the
equation by 8. Here we “undo” multiplication
by doing the opposite – division.
8n = 32
8
8
n=4
Example 4:
Solving a division equation
x
 11
9
To eliminate the 9 multiply both sides of the
equation by 9. Here we “undo” division by doing
the opposite – multiplication.


x
 11
9
x
9  (11)(9)
9

x  99
Identify operations
Undo operations
Balance equation
Repeat steps
Solve for variable
Check solution
Identify Operations
Minus sign means subtraction
x
38
2
Fraction bar means division
Use Opposite Operations
or “undo” Operations
Addition is opposite of subtraction (addition
undoes subtraction)
Subtraction is opposite of addition (subtraction
undoes addition)
Multiplication is opposite of division
(multiplication undoes division)
Division is opposite of multiplication (division
undoes multiplication)
Keep Equation Balanced
What ever you do to one side of the equation
you do to the other side of the equation.
Repeat these steps until the equation is solved.
Example:
7x + 15 = 85
7x +15 – 15 = 85 - 15
7x = 70
7
7
x = 10
Example:
2
x  6  28
3

2
x  6  6  28  6
3
2
x  28
3
3 2
3
x  28
2  3
2 


x  42
Graphing a Linear Equation
When graphing the solution to a linear equation with onevariable on a number line you would put a dot (point) on the
answer.
x – 3 = -7
x – 3 + 3 = -7 + 3
x = -4