Beware of the Sucker Answer
Make sure you answer
the question that is asked!
Double check the question
before you fill in the bubble!!
Do the Easy Ones First
Then go Back and
do the Hard Ones!
For Multiple Choice Tests
•Check each answer – if impossible or silly cross it out.
•Back plug (substitute) –
one of them has to be the answer
•For factoring – Work the problem backwards
•Sketch a picture
•Graph the points
•Use the y= function on calculator to match graphs
Read the
directions for
the test
carefully.
Read each
question
carefully.
Number next to a Letter (variable) means Multiply
4x
12a + -2b
If x = -3
If a = 5, b = -3
Then substitute &
multiply
Then substitute &
multiply
4(-3) = -12
12(5) + -2(-3) = 66
So do numbers next to ( ) and letters next to letters
x·y = xy
4(x) = 4x
a·b = ab
(-2)(a) = -2a
4·f ·g = 4fg
Addition and Subtraction are snobs. They just
combine with their own kind. They form cliques.
3x – 3a + 7 + 6x + 2a = 9x – a + 7
4x² + 6x +9 – 15x + 3x² + 10 = 7x² - 9x + 19
That’s Just How They Do. That’s How It Is.
Deal With It!
Multiplication and Division are party animals. They
will do it with anyone!
2x · 4y = 8xy
a· b·c = abc
12xy/4x = 3y
Understanding Algebraic Culture is the Key to Success!!
2=2÷3
3
Numbers come with Signs (+ -)
The sign is in front
Remember: These are the same thing
5–3=2
5+-3=2
Because Subtraction is Adding the Opposite
If you are confused, Circle the number & the
sign in front, then do the math!
2 – 56 + 7 – 8 – 10 = - 65
ABSOLUTE VALUE
The absolute value is always positive.
The absolute value of │5│ is 5
The absolute value of │-5│ is 5
To solve, drop the bars and make the inside
number positive
Watch Out! -│6│ = - 6
because the negative is lurking outside the bars.
ALGEBRA OPPOSITES
Opposite of Multiplication is
Division
25 * 4 = 100
100 ÷ 4 = 25
Opposite of Squaring is Square Rooting
= 25
=5
Algebra Truths
Adding Negative numbers - Think
MONEY $$$$$$, You won’t miss it.
No Equal Sign????? Simplify It!!
Combine Like Terms.
Simplifying is like cleaning up your room, put all the FROGS (Variables) together and all
the SMILEY FACES (Numbers) together, then do the Math.
Don’t Forget Numbers come with Signs!
4x -5 +6x +21 -8x
4x
+6x
-5
+21
-8x
--------------------------------------------------------------------------------------------------------------------------------
4x +6x -8x -5 +21
4x
+6x
-8x
-5
+21
--------------------------------------------------------------------------------------------------------------------------------
2x +16
2x
+16
You can’t add FROGS & SMILEY FACES, so
you are finished. Good Job!
Adding or Subtracting
**If signs are the same, add them and use the sign.
45 + 34 = 79, - 45 – 10 = -55
WATCH
YOUR
SIGNS
Negative number digging the hole.
Think Holes
Negative is cooling off.
Positive - dirt in
the hole.
Positive is warming up.
**If signs are different, subtract and use sign of larger number
-18 + 8 = -10, 60 – 20 = 40
Think temperature
–Positive number,
dollars in your
pocket.
Negative number,
dollars borrowed
Think money
WATCH YOUR SIGNS
Multiplying or Dividing
**If signs are the same, answer is positive.
4 * 8 = 32, -63 / -7 = -9
**If signs are different, answer is negative.
-6 * 7 = -42, -100 / 10 = -10
Distribution Principle
Multiply everything in parenthesis by number next
to parenthesis FIRST THING!!
THINK BIG MOUTH
SHOUTING “DO ME,
DO ME FIRST!!”
EXAMPLE : 3( 6x – 3)
3
(6x – 3)
= 18x - 9
Distribution Principle
Multiply everything in parenthesis by number next
to parenthesis FIRST THING!!
EXAMPLE :
Get them off the bus
So they can play
football.
3
3( 6x – 3)
6x– 3
18x - 9
Equation Solving???
Example:
2(10x – 3) = 6x + 2
Get the Teams off the Bus
Line up the Teams
Penalty for off-sides – must change signs
Huddle up
Man on Man Defense
Line of
Scrimmage
2(10x – 3) = 6x + 2
20x – 6 = 6x + 8
20x – 6x = 8 + 6
14x = 14
14x = 14
14 14
X=1
Think Football – Letters Vs. Numbers
≤ , ›, or ‹, or
≥
If the equation has an Inequality sign, follow the steps for
solving equations with = signs.
Play Football
Letters Vrs. Numbers
Last Play of the Game
If you have to MULTIPLY or DIVIDE by a negative
number, Be sure to FLIP the Inequality.
*****NOTICE*****
UGLY NUMBERS WORK THE SAME AS PRETTY NUMBERS!!!
UGLY Numbers Work the Same as PRETTY Numbers.
If you can solve:
2X + 10 = 40
Play Football
Letters Vrs. Numbers
Then you can solve:
5x + 1.93 = 4.56
And you can solve:
.5x + ⅝ = ⅓
Clue Words for writing equations from word problems
Word Clue
Word Sentence
Algebraic
Plus
1 plus 5
1+5
added to
6 is added to a
number
6+x
the sum of
The sum of 5 and
a number
5+x
A number is
increased by 10
x+10
15 is more than a
number
x+15
increasing
by
more than
Clue Words for writing equations from word problems
Word Clue
Word Sentence
Algebraic
Minus
6 minus 5
6-5
subtracted
from
7 subtracted from
a number
x-7
the
difference of
The difference of a
number and 10
x-10
decreased
by
A number is
decreased by 20
x-20
Less
5 less a number
5-x
less than
6 less than a
number
x-6
Clue Words for writing equations from word problems
Word Clue
Word Sentence
Algebraic
Times
7 times a number
7*x = 7x
Product
Product of 8 and a
number
8x
Doubled
A number doubled
2x
Twice
Twice a number
2*x = 2x
Of (fractions
and percents)
1/2 a number
1/2x
55% of a number
0.55x
Clue Words for writing equations from word problems
Word Clue
Word Sentence
Algebraic
Quotient
The quotient of a
number and 7
x÷7
divided by
10 divided by a
number
10÷x
*The first
number written before the clue word
will be the numerator
1. Consistent – one or many solutions
2. Inconsistent – No solution
1. Independent – Only one solution
2. Dependent – Has infinitely many solutions.
Slope
Y – Int
Graph
Type of
Systems
# of
Solutions
Same
Same
Same line
Consistent
Dependent
Infinitely
many
Same
Different
Parallel
Inconsistent
0
Different
Different/
Same
Intersects
Consistent
Independent
1
Slope – Intercept Form
y = mx + b
Slope- directions
Rise
Run
Y Intercept –
where to start
It’s a line address
If the slope is a whole number, put it on a stick m = 2
To Graph:
Example 1
slope is 2/1
Example 2
y = -3X+ 0
y = 2X + 1
y = -3X
Starts at 1
Starts at 0
Rise/run = 2/1
rise/run = 3/-1
Directions are up 2, over 1
Directions are up 3, over -1
Thanks to http://www.mathsisfun.com/equation_of_line.html
Linear Equations, Standard Form ax + by = c
Solving for y, Just 3 easy steps
Greenisms
Math Terms
1.Move x term(Change sides, Change signs)
2.Give x side a hug
3.Divide by number next to the y
Add/Subtract x term
Parenthesis ( )
Coefficient
Example: Solve for Y
2x – 7y = 12
Just 3 easy steps
1. -7y = 12 – 2x
(Move x term (Change sides, Change signs)
2. -7y = (12-2x)
(Give it a hug)
3. y = (12-2x) / -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS!!
Linear Equations, Standard Form ax + by = c
Solving for y, It’s a Football Game
Y VS Everybody Else
Follow football rules
Example: Solve for Y
2x – 7y = 12
Play Football
Y vs everybody else
Just 3 easy steps
1. -7y = 12 – 2x
X is offside, Penalty change signs
2. -7y = (12-2x)
Huddle up ( )
3. y = (12-2x) / -7 Man on man defense
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS!!
y2 – y1
x2 – x1
Slopes are Negative
Reciprocal
(Flip & Change Sign)
l lines
or
slope
y = mx + b
Find Equation of the Line:
y = mx + b
I need slope (m)
& the y-intercept
(b)
To find m – Solve the
equation for y and
use m
or use the
y –y
x – x formula
2
1
2
1
To find b - Plug x, y and
m into the line equation
and solve for b.
MY ANSWER:
y=
x +
(Just follow the Rules)
Exponent
Base
x²
Like Bases
Exponent
Multiply
Add
(x²a³g)(xa²g³) = (x³ a5 g )
Power up
Multiply
( a5 g³ y) ³ = a15 g 9 y3
Divide
Subtract
s
a³k
a6 x 4 k 5
a3 x7k 4 = x³
Add
No change
3a² + 5a² = 8a²
a² + a5 = a² + a5
Subtract
No change
10a² - 4a² = 6a²
Example
4
x
Example
5 4 3
x gk
2

2
4
ax k
1
5 = 5
x
Switch
Simplify
g
g x
2
3
7
2
5
4
3
axk k a xk
4 2
4
X
X²
X ∙ X = X²
X
X times X is X squared
Quadratics
Multiplying Binomials – Draw the Face
(x + 8) (x – 6)
Multiply
Watch your Signs
x² +8x
-6x
-48
Simplify (Combine Like Terms, eat the buggers)
x² + 2x -48
Check This
First
The Math “F” Word “Factoring”
1. Is there a common factor (number or letter)?
Yes
Proceed with CGF
NO.
Example
8x²y + 4x³ - 12x
Proceed to question 2
Factor each term
8x²y
2 * 2 * 2* x * x * y
4x³
2*2*x*x*x
12x
2*2*3*x
Circle common terms
8x²y
2 * 2 * 2* x * x * y
4x³
2*2*x*x*x
Multiply circled numbers
That’s your Common Factor
12x
2*2*3*x
Multiply leftovers, put in (
4x ( 2xy + x² -3 )
)
2. Perfect squares on end & 3 terms ?
NO, proceed
to question 3
x² - 10x + 25
YES, SPLIT IT NICE
x² - 10x + 25
Put out baggies to
hold the answer.
(parenthesis)
Split the first term
nice
(x - 5)(x - 5)
Sign between is
same as middle term
Place in baggies
ANSWER
(x -5)²
Split the second
term nice.
Place in baggies
3. Perfect squares on end & 2 terms?
Yes,
Same as question 2
Except the signs
are +,-
NO. proceed to
question 4
Example
x² - 64
Answer looks like this!
( x - 8 ) ( x + 8)
4. No perfect squares on end, 3 terms & starts with x²?
YES, its
quadratics in the
morning
NO, proceed to
question 5
A
M
x² - 10x + 24
Multiply to +24
Add to -10
3,8
11
-3, -8
-11
-2, -12
-14
-6,-4
-10
Put in the parenthesis
(x-6) (x-4)
Found it
All Done!!
5. Is there a number in front of the x² & does it have 3 terms
Example
2x² + 7x + 3
Yes, Jail Break
NO!
Then it is
Prime
1. Steal the “a” and give it to the last term (Multiply)
1
x² + 7x + 6
(2*3)
(can’t factor)
2. Search and Seizure (quadratics in the morning)(See question 4)
( x+6 )( x+1)
3. Arrested and Caught - Divide last terms by “a”
( x + 6/2 ) ( x + 1/2 )
4. Beat it Down - (reduce fractions)
All Done!!
( x + 3 ) ( x + 1/2 )
(x + 3 ) ( 2 x + 1 )
5. Parole (kick denominator to the front)
The Math “F” Word “Factoring” Summary
Check This First
Is there a common factor (number or
letter)?
Is there a Square on each End??
Greatest Common Factor (GCF)
Perfect Squares
1.
2.
3.
4.
5.
6.
7.
Example
Factor each term
Circle common terms
Multiply common terms
Write it down
Put out baggie for leftover
Multiply leftovers for each
term
Put in baggie ( )
8x²y + 4x³ - 12x
4x ( 2xy + x² - 3 )
1. Put out Parenthesis
2. Split FIRST and LAST
numbers nice
3. Put in Signs
3 Different Kinds
A. x² - 16x + 64
(x– 8) (x – 8)
B. x² + 18x + 81
(x + 9) (x + 9)
C. x² - 36
(x + 6) (x – 6)
The Math “F” Word “Factoring” Summary
No perfect squares and 3 terms
and starts with x²?
Is there a number in front of the x²
and does it have 3 terms?
Quadratics in the Morning (AM)
Jail Break
1. Make a factor tree
2. Multiply to last number, add
to middle number
3. Put out baggies (parenthesis)
4. Split first term nice
5. Drop in factors from tree
1.Steal the “a” and give it to the last
term (multiply)
2.Search and Seizure (Quad in AM)
3.Arrested and Caught (divide by “a”)
4.Beat Down (reduce fractions)
5.Parole (kick denominator to the
front)
6.Check it out. (FACE it)
Example
2x² + 7x + 3
Example
x² + 12x + 32
(x + 8) (x + 4)
1x² + 7x + 6
2(x + 6) (x + 1)
3(x + 6/2) (x + 1/2)
4(x + 3) (x +1/2)
5(x + 3) (2x + 1)
Systems of Equations
To Solve by Graphing
Solve with Graphing Calculator
1)
Make an x, y Chart
1)
2)
Select any x, solve for y
x 2x – 4
y
0 2(0) – 4 -4
1 2(1) – 4 -2
3) Then graph the two points.
4) Do for both equations.
5) The answer is where they
cross.
6) Be sure answer is in (x, y)
form
Solve each equation for y
(3 easy steps)
2)
Use y=
Y= button and enter
each equation
3) Use graph to eyeball answer
Or
2) Use 2nd TABLE to find
where y1 and y2 are equal
3) Be sure answer is in (x, y)
form
Systems of Equations
Solve by Substitution
(box & shove)
1) Solve one equation for x
or y (change sides, change signs)
2) Box it
3) Rewrite other equation
and shove box in
4) Solve for surviving letter
1) Distribute
2) Combine like terms
3) Solve
4) Send it back to box
5) Solve for other letter
6) Answer in (x,y) form
3y -2x = 11
y + 2x = 9
• y + 2x = 9
• y = 9 – 2x
• 3(9 - 2x) – 2x = 11
•
•
•
•
•
•
•
27-6x – 2x = 11
27 – 8x = 11
- 8x = -16
x=2
y = 9 – 2 (2)
y=5
(2, 5)
Systems of Equations
Solve by Elimination
2x – y = 9
3x + 4y = -14
1)
Look for opposite signs
• 2x – y = 9
3x + 4y = -14
2)
Multiply to create opposites
• 4(2x – y = 9)
3)
Add old and new equation
together
4)
Solve for surviving letter
• 8x – 4y = 36
• 3x + 4y = -14
• 11x
= 22
•
x=2
5)
6)
Plug back into either
equation (pick the easy
one)
Solve for other letter
• 2(2) – y = 9
• -y = 5
• y = -5
7)
Answer in (x,y) format
• (2, -5)