Download Chapter 2 Study Guide Answer Key

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Chapter 2 Study Guide
6
th
Grade A " E
Equations
An equation is a mathematical sentence stating 2 expressions are equal in value.
Equations can be: a) True
Ex: 2 + 3(4) = 14
b) False
Ex: 8 2 = 16
c) Open
Ex: 3x = 9
An open equation can be true or false depending on what you replace for the variable(s).
When solving an open equation, your goal is to find all possible values that can replace
the variable(s) to make the equation true. These are called solutions.
* Are the following true, false,
20.;. (4)(5) = 1 FQ'~e.
S(,") -= I
Solving Equations
a)
~
or open?
b) 8n + 2n
= 10n
101\ :
IOta./"
'-­
Tr~e..
c) n - 2 = 6
Optt\
"Solving" an equation involves a series of steps very important to the algebraic process. The process' ultimate goal is to get the variable alone, therefore providing a solution(s). 1- Simplify wherever possible before you start. 2 - Identify what operation and number is attached to the variable. 2 - Perform the inverse operation (+/- or xl.;.) to undo what is attached to the variable. 3 - Use the Golden Rule of Algebra to keep the equation "balanced". 4 - Perform a check to validate your solution makes the original equation true. Ex:
n - 7 = -3(5)
n\7=-15
+7
n =-8
+\
= -3(5)
-15 = -15 ./
Check: -8 - 7
Need to simplify -3(5)
Notice "subtract 7 is attached to In'. Inverse is "add 7".
Inverse on left, Golden Rule on right
Now 'n' is alone so we have solution form. Time to check.
H
In a check, show that the solution has been substituted
and evaluate both sides to make sure they're equal
Writing an equation involves translating from verbal to a math sentence.
Ex: The product of 7 and -6 is equal to 2 less than a number . Step 1: Assign a variable
Let n = a number Step 2: Re-read and understand 7(-6) = n - 2 Step 3: Write an equation that
matches the verbal phrase Step 4: Solve and show all steps
-42 = n - 2 +2
+2 -40 = n 7(-6) = -40 - 2 -42 = -42-/ Step 5: Check
*Solve for n. Show all work and complete a check.
d)
=4 (-3 - 5)
:. 'tl-t)
n - (-12)
rl ....ll.
n
e)
-5n
"
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:t
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m::
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...
f) -
n
= 4(- 6) - 3
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-s
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......., ~.z :: 'f(-8)
-~2. ':. -l2 y"
=-6 + ~2)
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o
~
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*Write an equation and solve.
g) The product of a number and -8 equals the sum of -30 and -10.
L~~
t\= Do,
I\"'~btr
-i('\=- ... "iO",-)O
~;"\
-6(5):: -JO"'-Io
":: -_'+0
-~
[~:.
h) C~'C~:
-'to:: -'to V
-S
5J
The quotient of a number and 4 is 3 more the product of -7 and 5
L~~ ~ -:: ,,~~~r
.!L :. -7 (5) ~"3
~
c..~,...""'.
~
= -7(S)-t~
4
-Sl.=- -1s- ... 3
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V
Coefficients and Terms Terms are mathematical values separated by addition or subtraction. A term can have a numeric factor and variable factors. The numerical factor is known as a coefficient. If no variable factors are present, the term is called a constant. 3x2y + 7x 2 + X - Y - 9 has 5 terms (3x~, 7x 2 , x, - y, -9)
Its coefficients are 3, 7, I, and -1 (x = Ix; -y = -ly)
-9 is a constant
It makes sense that -9 is called a "constant" since all the other terms' values depend on
how you replace the variable, but the term -9 "constantly" has the value of -9.
Ex:
Like terms are terms that have the same exact variable factors.
Ex:
3x~ + 7x 2 + X - Y- 9 has no like terms.
Yes, x~ and x 2 and x all have 'x' but none are exactly the same!!!
Ex:
lOx - 3xy - x + 2yx + xy2 have lOx and -x as like terms since both have
exactly x as a variable factor.
-3xy and 2yx are also like terms. Due to
Commutativity - xy and yx are the same.
*For the following, find: a) # of terms, b) coefficients, c) constants, d) like terms
i)
at) Lt
20x 3
""~(~S
-
x2 + 14x + 9 b'\
, J '20 J. -1 J. )'t
j)
6mp - 9m
+
p - 2pm - 4m - 3
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b) ,-. -9·J I I. -2'-Lf
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at)
, ""P Q"ef - 2 p~
-q M
~~
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Properties of Real Numbers
ab = ba
Commutative Property of Addition/Multiplication: a + b = b + a
changes order ." think "commute"
Associative Property of Addition/Multiplication: a + (b + c) = (a + b) + c
a(bc) = (ab)c
changes grouping .., think "associate"
Distributive Property of Multiplication over Addition: a (b + c) = ab + ac
Distributive Property of Multiplication over Subtraction: a (b - c) = ab - ac
a + 0 =a
Additive Identity Property:
a (1) =a
Multiplicative Identity Property:
Combining Like Terms
When like terms are in an expression, they must be combined. The Distributive Property justifies combining like terms. To combine like terms, add the coefficients and keep the variable factors the same. Ex:
3xy + xy = 3xy + lxy Multiplicative Identity
(3 + l)xy Distributive Property of Multliplication Over Addition
4xy
Simplifying Algebraic Expressions
An expression is fully simplified once:
a) all parentheses have been removed (ex: 2(3n) = 6n 4(x + 3) = 4x + 12)
b) all like terms have been combined (ex: 3x + 2 + x = 4x + 2)
c) all numerical operations have been evaluated (ex: 3 + 2y - 8 = 2y - 5)
Ex:
9m + (4 + 7m)
=
9m + (7m + 4) Commutative Property Addition
(9m + 7m) + 4 Associative Property Addition
(9 + 7)m + 4 Distributive Property Multiplication over Addition
*Simplify and justify each step.
k)
I)
12y + 2 + Y
2 (8q) + 4 ( m + q)
m) 5r + 3mr + 7
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Solve the following multi -step equations. Show all work. Include a check.
7\' - 1
n)
8 - 5x = -27
-S)(~8
-a
0)
p )
- 5
:: . . 27
-8
3-2(-3x-4)+2(-7x)=27
3+-2(-3)( -'t)
(~
:: -\I (-5)
r
~l
-+ ,
'"'"
I
-8 x 1-l\
C 'neck. ~
- -
&- 35 =-27 -21 =.-2.7'/
g
'2.7
- tI
~~:: ~
7
[X ~
%- S(7)~ -1.7 ~
....\ \
7>;.-:::. Sb
7
5:.... ,1:.(,
f:r)+
2(-7)t)~ 17
3 1" bX +ca - I'+x .:: 27
L
7)(,\-1 :: SS
of
=a\
1
-g
1)(
::-lJ
C\..~c:Jc
.....
7(&).:.' -If .:: -I S
-s
5{,-I
- - -'-I:: -,s
-s­
75s
-"f
~ -IS"
-II-'t -::: -IS'
"'J5~-'S./
3 - '( ( -S(-~)-'t) ~ ( (-7 (-l))
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... <. (I~) ~7.. 7
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~-«<.).,. ~~ ~27
3""-y.
of.
== 1.7
'2 ~
-.., +21> -::
27 -=-
::- '2 7
27
7.7/