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Algebra 1 Review
By Thomas Siwula
Addition Property (of Equality)
Example
a=b, then a+c = b+c
Multiplication Property (of Equality)
multiply the same number to each side
Example:
a=b then ac=bc
Reflexive Property
If one number equals itself
Example:
a=a
Symmetric Property
The numbers flipped around still equal each other
Example:
If a=b then b=a
Transitive Property
If one number equals a number that equals a third number
the first number is equal to the third number
Example:
If a=b and c=b then a=c
Associative
Property
of
Addition
If the parentheses switch the outcome is still the same
Example:
(a+b) +c = a+(b+c)
Associative Property of
Multiplication
Example:
(-3 x 7)x 5 = -3x (7 x 5)
Commutative Property of Addition
The order of the addition is switched and the sum is still the same
Example:
a+b=b+a
Commutative Property of
Multiplication
The order of the numbers being multiplied is switched
Example:
axb=bxa
Distributive Property (of
Multiplication over Addition
Multiply the numbers in the parentheses by the number outside of the
parentheses.
Example:
7(2+3)= 14+21
The seven was distributed to the 2 and 3
Prop of Opposites or Inverse
Property of Addition
A number plus its opposite is equal to zero
Example:
3+(-3)=0
Prop of Reciprocals or Inverse
Prop. of Multiplication
Example:
1
2
x
=1
2
1
Identity Property of Addition
A number plus zero equals itself
Example:
4+0=4
Identity Property of Multiplication
A number times one equals itself
Example:
5x1=5
Multiplicative Property of Zero
Any number times zero equals zero
Example:
13x0=0
Closure Property of Addition
If two real numbers are added together their sum will be a real number
Example:
The real numbers 10+9=19, another real number
Closure Property of Multiplication
Example:
The real numbers 4x6=24, another real
number
Product of Powers Property
Example:
n3xn4=n7
Power of a Product Property
Example:
(RS)11=R11S11
Power of a Power Property
Example:
(p7)3=p21
Quotient of Powers Property
Example:
w9/w6=w3
Power of a Quotient Property
Example:
49
2
(7/3) =
9
Zero Power Property
Anything to the power of zero is 1
Example:
(13)0=1
Negative Power Property
Anything put to the negative power is put under 1 to
make positive
Example:
7-2= 7
2
1
Zero Product Property
If one variable in the equation equals zero
the equation will equal zero
Example:
(n-10) (n-7)=0, therefore n-10
equals 0 or n-7 equals zero
Product of Roots Property
Example:
√28=√4√7
Quotient of Roots Property
Example:
√64/√4=√16=4
Power of a Root Property
Example:
52=25
Quiz
Look at the sample problem and give
the name of the property illustrated.
Click when you’re ready to see the answer.
1. a=b then ac=bc
Answer:
Multiplication Power of Equality
Quiz
Look at the sample problem and give
the name of the property illustrated.
Click when you’re ready to see the answer.
1. 4+0=4
Answer:
Identity Property of Addition
Quiz
Look at the sample problem and give
the name of the property illustrated.
Click when you’re ready to see the answer.
1. 3+(-3)=0
Answer:
Prop of Opposites or Inverse
Property of Addition
Quiz
Look at the sample problem and give
the name of the property illustrated.
Click when you’re ready to see the answer.
1. 13x0=0
Answer:
Multiplicative Property of Zero
Quiz
Look at the sample problem and give
the name of the property illustrated.
Click when you’re ready to see the answer.
1. 7(2+3)=14+21
Answer:
Distributive Property (of
Multiplication over Addition
Quiz
Look at the sample problem and give
the name of the property illustrated.
Click when you’re ready to see the answer.
1. The real numbers 10+9=19, which is another real number
Answer:
Closure Property of Addition
Quiz
Look at the sample problem and give
the name of the property illustrated.
Click when you’re ready to see the answer.
1. If a=b, then a+c = b+c
Answer:
Addition Property of Equality
Quiz
Look at the sample problem and give
the name of the property illustrated.
Click when you’re ready to see the answer.
2
x =1
1
Answer:
Prop of Reciprocals or Inverse
Prop. of Multiplication
1
1.
2
Quiz
Look at the sample problem and give
the name of the property illustrated.
Click when you’re ready to see the answer.
1. 3+5=5+3
Answer:
Commutative Property of
Addition
Quiz
Look at the sample problem and give
the name of the property illustrated.
Click when you’re ready to see the answer.
1. The real numbers 4x6=24, another real
number
Answer:
Closure Property of Multiplication
Quiz
Look at the sample problem and give
the name of the property illustrated.
Click when you’re ready to see the answer.
1. a=a
Answer:
Reflexive Property
Quiz
Look at the sample problem and give
the name of the property illustrated.
Click when you’re ready to see the answer.
1. a=a
Answer: Product of Roots Property
Quiz
Look at the sample problem and give
the name of the property illustrated.
Click when you’re ready to see the answer.
1. (a+b) +c = a+(b+c)
Answer:
Associative Property of Addition
Quiz
Look at the sample problem and give
the name of the property illustrated.
Click when you’re ready to see the answer.
1. (-3 x 7)x 5 = -3x (7 x 5)
Answer:
Associative Property of
Multiplication
Quiz
Look at the sample problem and give
the name of the property illustrated.
Click when you’re ready to see the answer.
1. AxB=BxA
Answer:
Commutative Property of
Multiplication
Quiz
Look at the sample problem and give
the name of the property illustrated.
Click when you’re ready to see the answer.
1. a=b b=a
Answer:
Symmetric Property
1st Power Equations
• These equations have one variable in them to the first power. Solve
by adding, subtracting, multiplying or dividing each side to get an
answer.
Ex: x+11=5, subtract eleven from each side to get x by itself and then x
= -6 is the answer.
Fractions with the 1st power
x x
Ex: 7  3  10, find the LCD, which in this case is 21. Multiply that to all
parts of the equation to cancel into- 3 x  7 x  210
Therefore 10x=210 and x= 21
-Equations with variable in the denominator2 3
Ex: 4a  5  24 ,find the LCD, which is 20a and multiply to all sides to
cancel into 10-12a=24, subtract 10 from each side to get,-12a=12,
divide and a=-1
Solving 1st Power Inequalities in One Variable
A. With one inequality sign- 5<X
B. Conjunction-3< X< 7
the word “and” can also be used in conjunctions
C. Disjunction - 4>X or 2-3<X
D. Special Cases
1. x<-3 and x>7 =  because x cannot be greater
than seven and less than -3 at the same time

Linear Equations in Two Variables
Slopes
A. Positive slope -Rises from bottom left to top right. Ex: y=3/2x+7, this has
positive slope because 3/2 is positive.
B. Negative slope-Otherwise known as falling lines and normally start at
top left
falls to bottom right. Ex: y= -5x + 2, this has a negative slope because
there
is a negative
C. Vertical slope- Occurs when y equals zero and x equals a number Ex:
x=3
The line will run vertically up and down the graph with a slope that is
undefined.
D. Horizontal slope- Occurs when x equals zero and y equals a number
Ex: y=9
The line runs horizontally across the graph and the slope equals zero.
-Graphing
In an equation such as y=3/2x+7, 7 is the y intercept so that would be
plotted on the y axis on the graph. From the point 7, since the slope is 3/2
one would count up three and over two to graph the linear equation. The
final product would look like this.
Linear Equations Continued
• There are two different types of slope form
1. Standard Form- Ax + By=C
2. Slope Intercept Form- Y=mx+b
Finding Slope and Intercept Points
Slope Formula- Gets the slope of the equation
Point Slope Formula- Once the slope is found, this formula
finds the y intercept, if it is unknown.
To find out the x intercept make y equal to zero
To find out the y intercept make x equal zero or use point
slope formula
Linear Systems
A. Substitution Method
Substitute an equation for a variable.
Ex: 9x+y=4, when y is isolated the equation is y=5x+4
-5x+3y=2, substitute 9x+4 for y, so the equation turns into 5x+3(5x+4)=2. This then is equivalent to -5x+15x+12=2, simplify and it is 10x=10,
where 1 is equal to x. Plug 1 into the first equation and y equals 9. The answer is
then (1,9)
B. Elimination Method
Eliminate one variable multiplying them by the LCF.
Ex: -3y-7x=6
7y+2x=10
Times 2x by 7 and -7x by 2 so they cancel eachother out
-3y-14x=6
7y+14x=10
-3y=6
7y=10
4y=16, therefore y=4 and then plug that into one of the original problems.
7(4)=2x=10, which simplified is 28+2x=10,
18=2x and x equals 9
The solution is (9,4)
Linear Systems Continued
•
After solving the linear equations and
graphing them, the lines will either be
1. Dependent-the equations both have the
same exact line. Dependent is also a
consistent line.
2. Consistent- There will be one point of
intersection between the two lines
3. Inconsistent-The lines are parallel and
will never intersect.
Ways to Factor
1. PST
2. GCF
3.Difference of Squares
4. Sum and Difference of Cubes
5.Reverse Foil
6.Grouping 2 by 2
7. Grouping 3 by 1
Rational expressions
A. Simplify by factor and cancel Factor the equation into conjugates and
then cancel the common factors. This leaves the equation in its simplest
form
Ex:
2  2 x  8 ( x  2)( x  4) x  2


x 2  9 x  20
( x  4)( x  5) x  5
x
B. Addition and subtraction of rational expressions, factor, find the LCD, and
multiply, add the numerators and cancel all common factors
Multiplication and division of
rational expressions
• For multiplication factor and cross cancel
• For division equations, flip the numerator
and denominator around to multiply
Ex: 3x2 - 4x
x(3x - 4)
3x - 4
-----------= -------------= ---------2x2 - x
x(2x - 1)
2x - 1
Quadratic Equations in One Variable
• Factoring
- Set the equation to zero, then factor.
Ex: 8x2- 40x=0
8x (x-5) = 0
8x = 0 and (x-5)=0
Solution x = (0, 5)
Ex: x2=36, set to zero, x2-36=0
(x-6) (x+6)= 0, therefore x = (-6,6)
Quadratics Continued
• Square Root of Both Sides
-Take the square root of the variable and the number
Ex: x4=25, take the squares of each side
4
x   25, 25 must have a plus minus sign in front of
it because x could equal + or – 5
x2= 5
Ex: x2=24, square, x2  24
Since 24 is not a perfect square you divide it into two
different squares, x   4 6 and then finish the problem
x  2 6
is the final answer
Quadratics
• Completing the Square
-Set a equal to 1 and ALWAYS put the equation into
standard form
Ex: 4x2+24=32, set to standard form
4x2+24-32=0, set a equal to 1
x2+6-8=0, now add c to the other side
x2+6 =8, add the equation ( )2 to bboth sides
x2+6+9=8+9, this is a PST, so factor 2
(x-3)2=17, take square roots from each side
( x  3) 2   17 simplify into,
x  3  17
Quadratics
b 
b 2  4ac
2a
• Quadratic FormulaWorks with the same equation, 4x2+24=32
Put into standard form, 4x2+24-32=0
Plug into formula and solve
b2-4ac is called the discriminate.
It is used to find out if a certain equation has
-two irrational roots-number is a non positive square
-two rational roots-when number is positive square
-one rational double root-when number is equal to zero
-no real roots-when number is negative
Functions
A. In a function, f(x) stands for y.
B. Finding Domain and Range
Domain(x)- set y, also known as the range to
zero, and then factor to find the domain
Range(y)-set x also known as the domain to
zero
C. When given two ordered pairs of data
such as (5,-8), (4,7) to see these points on a
graph
1. Use the slope formula to get the slope
2. Use the point slope formula to find the y
intercept and then graph the equation.
Quadratic functions
• When graphing a quadratic function the
graph will be a parabola.
• If a is negative the parabola opens
downwards, if a is positive the parabola
opens upwards.
Find the x and y intercepts by the means
that were stated in the previous slide.
Vertex equation- {-b/2(x)}
Axis of Symmetry- The axis = whatever the
vertex is
Graphing A Parabola
• Ex: f(x)=x2+6x+9
X intercepts- 0=x2+6x+9
Factor (x+3) (x+3), x intercepts= (-3,0)
Y intercept- y=0-0+9, y intercept = (0,9)
Vertex (-6/2x1), -3 = vertex
- Plug -3 back into equation
f(-3)=9-18+9=0
Axis of Symmetry = 0
Simplifying expressions with
exponent
Ex: 1. n10xn3=n13
2. 38/33= 35 or 33/38=1/35
3. (11n2p5)3= 33n6p15
4.-(72a3b2c-4)0=-1
5.x-3/5 - Flip the equation
around to make
exponent positive
Answer is 5/x3
Simplifying expressions with
radicals
-Radical expressions are square roots that are not
perfect so they need to be broken down before any
squares can be found
Ex. √24=√4√6=2√6
Ex: √320=√64x5=4√5
Word Problems1. A model airplane is propelled upward with a start speed of 36 ft/s. After how many seconds does it
return to the ground? Plug the data into the equation h =rt-16t2 , where h is height, r is rate, and t is
time.
The starting equation will look like this- h=36t-16t2
Solve for t by means of GCF and factoring
2. In 3 days Jane lost 8 pounds, and then in 9 days Jane
had lost 20 pounds. If the growth continues linearly, write an
equation Jane could use to predict her weight on day 9.
(Hint: Use the slope formula and the point slope formula to
help with an answer.)
To solve this plug in the data to the point slope and slope
formula to make an equation that would solve the problem.
3. A jar contains 19 coins in quarters and dimes, if the total
value of the coins is 2.85, how many of each coin is there?
To solve make variables for quarters and dimes. Then make
the variables added together like this q+d=19. Plug the
variables into this equation.
Word Problems
•
3. A jar contains 19 coins in quarters and dimes, if the total value of the coins is 2.85,
how many of each coin is there?
To solve make variables for quarters and dimes. Then make the variables added
together like this, q+d=19 and isolate one variable, like so d=19-q. Plug the variables
into this equation. 25q+10d=285 and then use the substitution method.
25q+10(19-q)=285. 25q+190-10q=285 Solve this equation solving for q and then plug q
back into the equation of d=19-q to solve for d.
4.The length of a rectangle is 6 more than twice the width. The perimeter is 94. Find the
dimensions of the triangle.
-Set the variable w for the width and formulate an equation.
-The equation would look like this 2(2w+6)+2w=96. The 2(2w+6) stands for the length of
each side of the rectangle and the rest of the equation, 2w is for the width.
Solve- 4w+12+2w=96
6w+12+96
6w= 84 and w = 14
Line of Best Fit or Regression Line
A. Line of Best Fit Regression is most commonly
used when predicting. Line of Best can predict the values of
a dependent variable when compared with the values of an
independent variable.
B. A calculator helps greatly with regression lines because
one can simply plug in dependent and independent data
into the calculator. The calculator will put that information
into a graph and create a regression line that is ideal for
making predictions of a certain variable.
Regression Problem
C. Try and find what the price of the house will be
in year 7 using line of best fit and your calculator
Years
House is
For Sale
1
2
3
4
5
Price
Decline
200,000
196,000
183,000
175,000
167,000