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Algebra-I
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A math & science tutorial center
Number System
Slopes & Lines
Real numbers
Irrational
Slope > 1
Slope < 1
5
π
Rational
Can expressed as fractions
Integers = -2, -1, 0, 1, 2
Whole=0, 1, 2
Natural= 1, 2,3
Slope = 0
Y = number
Slope = m =
Slope undefined
X =number
y2-y1
_____
x2-x1
Slope-Intercept Form
y = mx + b
m = slope
b = y-intercept
Properties of Operations
Commutative
addition
Multiplication
a+b=b+a
axb=bxa
Associative (a+b)+c = a+(b+c)
(ab)c = a(bc)
Point-Slope form
y-y1 = m(x-x1)
(x1,y1) = y-intercept
Identity
ax1=a
Standard Form
Ax + By = C
Distributive
X-intercept
To find x-intercept, y=0
Y-intercept
To find y-intercept, x=0
Inverse
a+0 = a
inverse of 5 is -5 inverse of 5 is 1/5
Properties of Equality & Inequality
equality
Reflexive
Intersecting
One solution
overlapping
Many solutions
inequality
a=a
Symmetric
If a = b,
then b =a
Transitive
If a=b & b=c,
then a = c
If a<b & b<c,
then a<c
Addition
If a=b, then
a+c = b+c
If a<b, then
a+c < b + c
Subtraction
If a=b, then
a–c=b–c
If a<b, then
a–c<b–c
Multiplication
If a=b, then
axc=bxc
If a<b, then
axc<bxc
Division
If a=b, then
a/c = b/c
If a<b, then
a/c < b/c
Number of Solutions
Parallel
No solution
a·(b+c)=(a·b)+(a·c)
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Algebra-I
FREE step-by-step solutions
Free additional copies & subjects
A math & science tutorial center
To find equation of line
through A (3, 7) & B (–3,2)
Note: basic equation is y = mx + b
1) Find slope using A and B
m=
y 2 − y1
= 7−2 = 5
x2 − x1 3 − (−3) 6
2) Re-write basic equation with actual m
y = 5x+b
6
Roots
Addition
3 5+4 5 =7 5
Subtraction
7 5 −4 5 =3 5
Product
2
15 × 25 = 2 15 × 2 25
3) Pick A or B to substitute in for x & y
In this example we will use B (–3,2)
2 = 5 (−3) + b
6
Quotient
4) Solve for b
Root of a root
b=9
2
5) Rewrite basic equation with numbers
y =5x+9
6
2
4
35
15 = 4 15
25 4 25
25 = 3×5 25 = 15 25
DONE
Factoring
Graphing lines
Using y = mx + b
1) Find b on the y-axis
2) Follow the slope by
up/down then right/left
3) Connect the points
Using X & Y-Intercept
1) Find x-intercept using y=0
2) Plot the x-intercept
3) Find y-intercept using x=0
4) Plot the y-intercept
5) Connect the points
Multiplication – add exponents
503 × 504 = 507
Division – subtract exponents
6
50 = 502
504
Exponent to an exponent – multiply
6 3
= 5018
Exponent to parenthesis
(15 × 8)3 = 153 × 83
(158 ) = 158
3
Difference of Squares:
(a - b)2 = a2 - 2ab + b2
Sum of Cubes:
a3 + b3 = (a + b) (a2 – ab + b2)
Difference of Cubes:
a3 - b3 = (a - b) (a2 + ab + b2)
Quadratic Formula
Exponents
(50 )
Perfect Squares:
(a + b)2 = a2 + 2ab + b2
3
3
Negative Exponent
25−2 = 12 = 1
625
25
Zero Exponent - always equals 1
Any number to the zero = 1
for example: 3450 = 1
–b±
x=
b2 – 4ac
2a
Discriminant
b2 – 4ac > 0, two real solutions
b2 – 4ac = 0, one real solution
b2 – 4ac < 0, no real solution
Distance Formula
d = ( x2 − x1)2 + (y 2 − y1)2
Midpoint Formula
xmp =
x2 + x1
2
;
y mp =
Pythagorean Theorem
a2 + b 2 = c 2
c = hypothenus
y 2 + y1
2