ALGEBRA 1, Sheet 1
See also Pre-Algebra, Algebra II, Geometry, Precal, Calculus Sheets
Adding & Subtracting
positive and negative numbers
SAME SIGN ADD
DIFFERENT SIGN SUBTRACT
Multiplying & Dividing
positive and negative numbers
SAME SIGN POSITIVE +
DIFFERENT SIGN NEGATIVE |Absolute Value|
DISTANCE from ZERO
ALWAYS POSITIVE +++
|8| = +8
|125| = +125
|- 8| = +8 The only possible |- 125| = +125
negative outcome
- |8| = - 8 is if it is out in
- | 125| = -125
FRONT of the
- | - 8| = - 8 absolute value - |- 125| = -125
Evaluate Algebraic Expressions
Plug in the given variables, solve!
2
Evaluate x - 5y for x = 2 and y = -1
2
x - 5y =
(2)2
- 5(-1) = 4 - (-5) = 9
Evaluate 3x2 + 2y for x = 5 and y = -4
3x2 + 2y= 3(5)2 + 2(-4) =75 - 8=67
Slope
m = y2 - y1
x2- x1
5 + 3 same sign add = 8
-5 - 3 same sign add = - 8
-5 + (-3) same sign add = - 8
-5 + 3 different sign subtract = - 2
-6 x -7 same sign + = + 42
-42 ÷ -7 same sign + = + 6
6 x -7 different sign - = - 42
42 ÷ -7 different sign - = - 6
-7 x 6 different sign - = - 42
42 ÷ -6 different sign - = - 7
Distributive Property
Order of Operations
PEMDAS
Please (Parentheses)
Multiplication and
Excuse (Exponents)
Division are done
left to right.
My (Multiplication)
5(x +1) = 5•x + 5•1 = 5x + 5
Dear
(Division)
2
y(2y - 3) = y•2y - y•(-3) = 2y - 3y
Addition and
Aunt (Addition)
Subtraction are
12(a2 + 5b)= 12•a2+12•5b = 12a2 + 60b Sally (Subtraction)
done left to right.
Distribute (or multiply) the term outside
the parentheses times EACH TERM
INSIDE THE PARENTHESES.
Adding and Subtracting Variables
Only add and subtract LIKE TERMS
Examples of “like terms”:
4x, -10x, 100x, -3x
4ab, -10ab, 100ab, -3ab
4y, -10y, 100y, -3y
4x2, -10x2, 100x2, -3x2
“like term”
x
ab
y
x2
4x - 10x = -6x
4x -10y = NOT POSSIBLE
4ab + 100ab = 104ab
4y - 10y + 100y = 94y
4x2 + 100x = NOT POSSIBLE
4x2 - 3x2 = 1x2
2x • 3x2 = 2 • 3 • x1+2 = 6x3
4
2 3
2
4+3
UNlike terms can be multiplied and divid- 5y • 6x y = 5 • 6 • x • y
b = y-intercept
ed. Multiply and divide whole numbers
separate of the variable.
3
2
3-2
1
Multiplying Fractions
Find a number that can be divided evenly in both numerator and denominator. Keep doing this until you
can no longer divide, that’s when it is simplified.
z
8
34 24
8
y
3
Exponents
ADD exponents, → multiplying
SUBTRACT exponents → dividing
C273 27.1 2 C2
5 83 x
t2a 4a
5 + -3 different sign subtract = 2
42 ÷ 7 same sign + = + 6
Simplifying Fractions
8 2.3
-5 + +3 diff sign subtract, larger number (-), ans (-)
6 x 7 same sign + = + 42
ADD exponents when multiplying
SUBTRACT exponents when ÷
I I
-5 - (-3) different sign subtract= - 2
(the sign on the bigger number = sign of answer)
Slope-Intercept Form
(x1, y1) (x2, y2)
y = mx + b
= given points
Point-Slope Form
y - y1 = m(x - x1)
5
5 - 3 different sign subtract = 2
(the sign on the bigger number = sign of answer)
(the sign on the bigger number = sign of answer)
Mulitplying and Dividing Variables
m = slope
5 - (-3) same sign add = 8
5 + +3 same sign add, equals positive 8
4a3
8
5
24
7
@CuteCalculus
4a ÷ a = 4a
5
240
2
g
Exponents Raised to Exponents
MULTIPLY exponents when raised to
another exponent.
302
16 6
oy5K83 y5.3 51255
4
3
FOR MORE HELP DM @CuteCalculus
47
Tutoring in Pre-Algebra, Algebra I & II, Geometry, Precal, Calculus - DM @CuteCalculus
= 4a
5-3
8x ÷ 2x = 4x
= 4x2
18x2y3 ÷ 3y= 6x2y3 - 1 = 6x2y2
Dividing Fractions
Multiply top • top, bottom • bottom.
5zx
3
=30x2y7
Flip the second fraction and multiply.
Eye
x5z
oq
x
3 4 17
561
4 41 159
NEGATIVE Exponents
If a term has a negative exponent and is in
the numerator, move it to the denominator to
become positive. If the term with the negative
exponent is in the denominator, move it to the
numerator to become positive.
t
s
s
I4Ia
j.az
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When solving equations, WHATEVER YOU DO ON ONE SIDE OF THE EQUATION, YOU MUST DO ON THE OTHER!
Algebra 1, Sheet 1
Graphing
2x 4
y
y4
2
i
i l
y
p
p
n
F
4
ti
Using slope intercept
y = mx + b
1
l l el nl t ii zl a1x
y 2x 4
Step 1, plot y-int = 1
Step 2, m=-1/3 (fall 1, run 3)
Step 3, plot at least 3 points
Step 4, draw the line
m 2
x y
1
y 3
b 1
y 31 1
4
Step 1, plot y-int = -4
Step 2, m=2 (rise 2, run 1)
Step 3, plot at least 3 points
Step 4, draw the line
y mxtb
m z
a
i
Write the equation of a line passing through
the point (-3, 7) with a slope of 2.
Write the equation of a line with a
slope of 3 and y-intercept of -1.
t
I k
Using point slope
y - y1 = m(x - x1)
f3,7
21
y 7
3
Graphing Horizontal and Vertical Lines
y = number is a HORIZONTAL LINE going
a thin y 3
through the given y-value. This is when the
s
a
slope is ZERO, therefore there is no x value.
z
ix
it s
x = number is a VERTICAL LINE going through 1 111
the given x-value. This is when the
yy
slope is UNDEFINED or DOES NOT EXIST.
Goal to get x by itself!
Solving Equations
Finding x-intercept
Finding y-intercept
Is
where
y
=
0,
solve.
Is
where x = 0, solve.
12
x
3
4 312 51 7
x 12 3
12 12
4 372
69 12 2X t 69 12
751 7 9 2X t 6101
a
2101 69 12
2
12
Add 12 to both
4 6 15 7 9
cop
can 69 12
sides.
z yz
10 15 7 9
s
s
6 6,0
y z
0,2
tx
tx
Ex 9 5
Quadratic Formula
x = the solution to the
3 15 9
I
quadratic equation.
is is
3 Ex g 3
a
These are also called
5 6
5 6
s s
x i
x
Subtract 5 from
both sides.
4
4
4
8 32
8 32
o_O To
4
x
Divide 8 from
both sides.
1. Distribute 3
2. Combine
like terms
3. Subtract 7x
4. Subtract 15
5. Divide by 3
45 15
Multiply by
reciprocal (5/3)
on both sides.
x z
@CuteCalculus
the “zeros”,
“x-intercepts”,
“where the function
crosses the x-axis”,
“the value of the
function when y = 0.”
Inequalities
3
z
3
Sx
Sx
5
s
3
F
z
Subtract 3 from
both sides.
x
7
3xc 25
SE 7
3xc 25
zo
20
5
4
7
F
Divide by -5 on
both sides.
7
<>≤≥
SYMBOL SWITCHES DIRECTION
when multiplying or dividing by
a negative number!
stept
IF
6
x
6a
4
as
Always write answer from least to greatest
Absolute Value Equations
Absolute value is the distance from zero, it is
always positive so you must account for the
possibility that the number inside the absolute
value could have been negative and therefore
get two possible answers.
11 6
x
6x
1 11 8
1 8 x 1
x i
3
6
8
x a
Once the absolute value is isolated,
write TWO equations, one written as
is and the other = to negative
6 4
16
6
2x 2
E
x I
y
3
6
6
zx
I
x
d = distance between two points.
6
x ba Xz.Yz
E
d VXz X 2t Yz y
s
Tutoring in Pre-Algebra, Algebra I & II, Geometry, Precal, Calculus - DM @CuteCalculus
ye 3x i
a
a
Step 1, plot y-int = -1
Step 2, m=- 2/3 (fall 2, run 3)
Step 3, solid line for ≤
Step 4, shade below for ≤
Step 1, plot y-int = 1
Step 2, m=2 (rise 2, run 1)
Step 3, dotted line - - - for >
Step 4, shade above for >
Distance Formula
yo
44
y
a
1
14 4 2 2,41242
61 2
2
a
tx
2
Ix
i
X ba Xz.Yz
7
12 61 4
2
i
2a
yE
Midpoint Formula
7
3 2
1
2
i
eh A
I
b Vb 49C
x 1
Graphing Inequalities
Midpoint = average of the two points.
6 7 5
3 2
y
d
t
t
43
y
x
Parallel lines =
ARE EQUAL = THE SAME!
y = 3x + 8 and y = 3x - 1
y = -1/4 x - 5 and y = -1/4 x
y = -x + 7 and y = -x - 9
y = 5x and y = 5x + 2
Perpendicular lines
Opposite sign, flipped fraction
y = 3x + 8 and y = -1/3x - 1
y = -1/4 x - 5 and y = 4x
2 y = -x + 7 and y = x - 9
y = 5x and y = -1/5x + 2
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