Transcript (-a) = 0

Algebra 1
Marcos De la Cruz
Algebra 1(6th period)
Ms.Hardtke
5/14/10
Algebra Topics
1- Properties
2- Linear Equations
3- Linear Systems
4- Solving 1st Power Equations (1 Variable)
5- Factoring
6- Rational Expressions
7- Quadratic Equations
8- Functions
9- Solving 1st Power Inequalities (1 Variable)
10- Word Problems
11- Extras
Addition Property of Equality

If the same number is
added to both sides of
an equation, both sides
will be and remain equal

3=3 (equation)



If 2 is added to both sides
2+3=3+2
5=5

Negative Special Case

y=3x+5 (equation)




If (-3) is added to both
sides
Y-3=3x+5-3
Y-3=3x+2
Its still equal
Multiplication Property of Equality

States that when both sides of an equal
equation is multiplied and the equation
remains equal

If 5=5 (equation)

5x2=2x5


You multiply both sides by 2
10=10

Still remains equal
Reflexive Property of Equality

When something is the
exact same on both
sides



A=A
7x = 7x
3456x = 3456x
Symmetric Property of Equality

When two variables are different but are the
same number/amount (equal symmetry)



If a=b, then b=a
If c=d, then d=c
If xyp=xyo, then xyo=xyp
Transitive Property of Equality

When numbers or variables are all equal



If a=b and b=c, then c=a
if 5x=100 and 100=4y, then 4y=5x
if 0=2x and 2x=78p, then 78p=0
Associative Property of Addition

The sum of a set of numbers is the same no
matter how the numbers are grouped.
Associative property of addition can be
summarized algebraically as:
(a + b) + c = a + (b + c)

(3 + 5) + 2 = 8 + 2 = 10


3 + (5 + 2) = 3 + 7 = 10
(3 + 5) + 2 = 3 + (5 + 2).
Associative Property of Multiplication

The product of a set of numbers is the same no
matter how the numbers are grouped. The
associative property of multiplication can be
summarized algebraically as:
(ab)c = a(bc)
Commutative Property of Addition

The sum of a group of numbers is the same
regardless of the order in which the numbers
are arranged. Algebraically, the commutative
property of addition states:
a+b=b+a


5 + 2 = 2 + 5 because 5 + 2 = 7 and 2 + 5 = 7
-3 + 11 = 11 - 3
Commutative Property of
Multiplication

The product of a group of numbers is the same
regardless of the order in which the numbers
are arranged. Algebraically, commutative
property of multiplication can be written as:
ab = ba

8x6 = 48 and 6x8 = 48: thus, 8x6 = 6x8
Distributive Property

The sum of two addends
multiplied by a number
is the sum of the
product of each addend
and the number

A(b+c)


Ab + Ac
3x(2y+4)

6xy + 12x
Property of Opposites/Inverse Property
of Addition

When a number is added by itself negative or
positive to make zero
a + (-a) = 0


5 + (-5) = 0
-3y + (3y) = 0
Property Of Reciprocals/Inverse
Property of Multiplication

For two ratios, if a/b =
c/d, then b/a = d/c



a(1/a) = 1
5(1/5) = 1
8/1 x 1/8 = 1


A number times its
reciprocal, always
equals one
A Reciprocal is its
reverse and opposite
(the signs switch from +
to — or vice versa)
Reciprocal Function (continued)

The reciprocal function: y = 1⁄x. For every x except 0,
y represents its multiplicative inverse
Identity Property of Addition

A number that can be added to any second
number without changing the second number.
Identity for addition is 0 (zero) since adding
zero to any number will give the number itself:
0+a=a+0=a


0 + (-3) = (-3) + 0 = -3
0+5=5+0=5
Identity Property of Multiplication

A number that can be multiplied by any second
number without changing the second number.
Identity for multiplication is "1,“ instead of 0,
because multiplying any number by 1 will not
change it.
ax1=1xa=a


(-3) x 1 = 1 x (-3) = -3
1x5=5x1=5
Multiplicative Property of Zero

Anything number or
variable multiplied
times zero (0), will
always equal zero
5 x 0=0
 5g x 0=0
No matter what number is
being multiplied by zero,
it will always be zero

A really long way to explain the Multiplicative Property of Zero
(Proof)
http://upload.wikimedia.org/math/b/5/8/b5892630f1d2f28a580331a1d7e3e79f.png
Closure Property of Addition

Sum (or difference) of 2 real numbers equals a real
number

10 – (5)= 5
Closure Property of Multiplication

Product (or quotient if denominator 0) of 2 Reals
equals a real number

5 x 2 = 10
(Exponents) Product of Powers
Property

Exponents
Exponents are the little
numbers above
numbers, that mean that
the number is multiplied
by itself that many times

7×7 =
2
(7 × 7)
× (76 × 7 × 7 × 7 × 7 × 7)

When two exponents or
numbers with exponents
are being multiplied,
you add both exponents,
but you still multiply the
number or variable

3x (5x ) =


3
15x 4
15x(3+4)
7
Power of a Product Property

To find a power of a product, find the power
of each factor and then multiply. In general:
(ab) = a · b
m
Or
m
m
a · b = (ab)
m

m
(3t)

4
(3t) = 3 · t = 81t
4
4
4
4
m
Power of a Power Property

To find a power of a power, multiply the
exponents. (Its basically the same as the
Power of a Product Property, if forgotten, go
one slide back and review.)

3 4
(5 )


3
3
3
3
(5 )(5 )(5 )(5 ) = 5
3(4)
5
b c
bc
Its basically this:
(a ) = a
12
Quotient of Powers Property


When both the denominator and
numerator of a fraction have a
common variable, it can be
canceled, therefore not usable
anymore
Also when a variable is canceled,
the exponents are subtracted,
instead of added as in the Product
of Powers Property


b
c
a /a
5
3
——
2
5
b-c
a
5x5x5
——————
5x5
=5
(the canceling of common
factors)
Power of a Quotient Property


This is almost the same as the Quotient of
Powers Property, but this time, an entire
fraction is multiplied by an exponent
You also have to cancel the common factors, if
there are any

c
(a/b) c
a /b
(and vice versa)
c
— (a/6)
2
(a /36)
2
Zero Power Property

If a variable has an exponent of zero, then it
must equal one




0
a =1
0
b =1
c 0 b 0a 0=1
2 0
(a ) =1
Negative Power Property

When a fraction or a number has negative
exponents, you must change it to its reciprocal
in order to turn the negative exponent into a
positive exponent
-2

4
¼
2
1/16
The exponent turned from negative to positive
Zero Product Property

When both variables equal zero, then one or
the other must equal zero



if ab=0, then either a=0 or b=0
if xy=0, then either x=0 or y=0
if abc=0, then either a=0, b=0, or c=0
Product of Roots Property

The product is the same as the product of
square roots
Quotient of Roots Property

The square root of the quotient is the same as
the quotient of the square roots:
A
A
B
B
Root of a Power Property
Power of a Root Property
Density Property of Rational Numbers

Between any two rational numbers, there
exists at least one additional rational number
1
2
3
4
5
4.5
or
4½
6
7
8
9
Websites
PROPERTIES
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http://www.my-ice.com/ClassroomResponse/1-6.htm
http://intermath.coe.uga.edu/dictnary/descript.asp?termID=300
http://en.wikipedia.org/wiki/Multiplicativeinverse
http://ask.reference.com/related/Reciprocal+of+a+Number?qsrc=2892&l=dir&o=10601
http://faculty.muhs.edu/hardtke/Alg1_Assignments.htm — MUHS
http://www.northstarmath.com/sitemap/MultiplicativeProperty.html
http://hotmath.com/hotmath_help/topics/product-of-powers-property.html
http://hotmath.com/hotmath_help/topics/power-of-a-product-property.html
http://hotmath.com/hotmath_help/topics/power-of-a-power-property.html
http://hotmath.com/hotmath_help/topics/quotient-of-powers-property.html
http://hotmath.com/hotmath_help/topics/power-of-a-quotient-property.html
http://hotmath.com/hotmath_help/topics/properties-of-square-roots.html
http://www.slideshare.net/misterlamb/notes-61
http://www.ecalc.com/math-help/worksheet/algebra-help/
Hotmath.com
Algebra Topics
1- Properties
2- Linear Equations
3- Linear Systems
4- Solving 1st Power Equations (1 Variable)
5- Factoring
6- Rational Expressions
7- Quadratic Equations
8- Functions
9- Solving 1st Power Inequalities (1 Variable)
10- Word Problems
11- Extras
Standard/General Form



Standard Form
Ax + By = C
The terms A, B, and C are
integers (could be either
positive or negative
numbers or fractions)
If Fractions:


Multiply each term in the
equation by its LCD (Lowest
Common Denominator)
Either add or subtract to get
either X or Y isolated, in one
side of the =

If Decimals:

Multiply each term in the
equation depending on the
decimal with the most
numbers (by 10, 100, 1000,
etc)

1.23 (multiply times 100)



123.00
Subtract or add to get X or Y
isolated
If Normal Numbers (neither
fractions or decimals):

Just add or subtract to get X
or Y isolated
Graph Points


A Graph Point contains of an X and a Y
(x,y)
The X and Y mean where exactly the point is
located
Y line graph
X line graph
Standard/General Form Ex.

Fractions:

You multiply by the
LCM


Which in this case is
20x
Then to double check
it…
Point-Slope Form

The Point-Slope form. got its
name because it uses a single
point in a graph and a on the slope
of the line

It is usually used to find the slope
of a graph, if the slope is not
given in a certain problem or
equation


The Y on the Point-Slope form.,
doesn’t mean that the Y is
1
multiplied by one, but it means to
use the first Y of the two or one
point given as a problem (same
with X)
ex
 (4,3) and the slope is 2
 M = slope
 Y—stays the same
 X1 —is 4 (because 4 is in
the x spot)
 Y1 — is 3
 X—stays the same
If the problem gives you two
points and no slope, then you are
free to choose what which or the
Xs or the Ys you may want to use
for your Point-Slope Form.
Point-Slope Form

(4,3) and m=2

you must convert “it” to a slope-intercept form
Y=Mx + B

Y-3 = 2(x-4)


Y-3 = 2x-8
Y = 2x – 11 (slope-intercept form)
Slope-Intercept Explanation




y=mx+b
Sometimes in the Slope-intercept form, there are
fractions as the slope or the y-intercept
B= y-intercept
Rise/Run



When the slope is a fraction, you mark the B in a graph,
which is the y-intercept
Then depending on the slope, if its positive than the line
will look like this…
If its not positive, but negative, it will look like:
Point-Slope (Slope-Intercept) Graph

Y = 2x – 11
Rise/Run
Go up twice and to the side once
(5,0)
(0,-11)
Websites
(for further information)
Linear Equations



http://www.algebralab.org/studyaids/studyaid.aspx?file=Algebra1_5-5.xml
http://www.freemathhelp.com/point-slope.html
http://www.wonderhowto.com/how-to-solve-mixed-equation-decimal-percent-fraction-303082/
Algebra Topics
1- Properties
2- Linear Equations
3- Linear Systems
4- Solving 1st Power Equations (1 Variable)
5- Factoring
6- Rational Expressions
7- Quadratic Equations
8- Functions
9- Solving 1st Power Inequalities (1 Variable)
10- Word Problems
11- Extras
Linear Systems— Method Explanation





Substitution
The Substitution Method, is used
when, there are two equations, and
you pick one (the one that looks the
easiest to do) and you isolate either
the x or the y
When x or y is isolated, then you will
get something like this:
 Y= ?x + ?
 X= ?y + ?
Then, you replace the x or the y in the
equation that you didn’t touch yet,
and you must insert
If you isolated the y, then you will
solve for x
If you isolated the x, then you will
solve for y
Elimination






The Elimination Method, is used
when there are two equations and, it
is said to be a lot easier than the
Substitution Method
First, you will have to decide whether
you want to go for the x or the y
Then, you will multiply and
cancel/eliminate either x or y
depending, on which one did you
chose to do (x or y)
Then you solve for x or y
You will eventually substitute, more
like insert your y or x answer into the
either problem replacing it with x or y
Then you solve for either x or y
Substitution Method
y = 11 - 4x
Isolate the Y or X
Substitute the number, insert it
x + 2(11 - 4x) = 8
Solve for X and Solve for Y (vice versa)
Answers
http://www.tpub.com/math1/13d.htm
Literal Coefficients
Simultaneous equations
with literal coefficients
and literal constants
may be solved for the
value of the variables
just as the other
equations discussed in
this chapter, with the
exception that the
solution will contain
literal numbers. For
example, find the
solution of the system:
3 Variables !!
We proceed as with any
other simultaneous linear
equation. Using the
addition method, we may
proceed as follows: To
eliminate the y term we
multiply the first equation
by 3 and the second
equation by -4. The
equations then become …
To eliminate x, we multiply
the first equation by 4 and
the second equation by -3.
The equations then
become
We may check in the same
manner as that used for
other equations, by
substituting these values
in the original equations.
Elimination Method


2x – 3y = 19
5x – 2y = 20




2x – 3y = 19 (2)
5x – 2y = 20 (-3)
4x – 6y = 38
-15x + 6y = -60

-11x




The two equations

Now we multiply and then
later cancel out a variable,
depending which one you
chose
Now we got one answer—x
=2
Now we must insert the two,
into the either of the
equations…(substitution
method)
Now you got the y = -5

= -22
X=2
2x – 3y = 19
2(2) – 3y = 19



4 – 3y = 19
-3y = 15

Y = -5

Dependent



When a system is "dependent," it means that ALL
points that work in one of them ALSO work in the
other one
Graphically, this means that one line is lying entirely
on top of the other one, so that if you graphed both,
you would really see only one line on the graph, since
they are imposed on top of each other
One of them totally DEPENDS on the other one
Independent




When a system is "independent," it means that they
are not lying on top of each other
There is EXACTLY ONE solution, and it is the point
of intersection of the two lines
It's as if that one point is "independent" of the others.
To sum up, a dependent system has INFINITELY
MANY solutions. An independent system has
EXACTLY ONE solution
Consistent



We say that a point is a "solution" to the
system when it makes BOTH equations true,
right?
This is to say that there exists a point (or set of
points) that "work" in one equation and also
"work" in the other one
So we say that this point is CONSISTENT
from one equation to the next
Inconsistent



On the other hand, if there are NO points that
work in both, then we say that the equations
are INCONSISTENT
NO numbers that work in one are consistent
with the other
To sum up, a consistent system has at least one
solution. An inconsistent system has NO
solution at all
Websites
Linear Systems:



http://www.tpub.com/math1/13d.htm
http://mathforum.org/library/drmath/view/62538.html
http://www.purplemath.com/modules/systlin2.htm
Algebra Topics
1- Properties
2- Linear Equations
3- Linear Systems
4- Solving 1st Power Equations (1 Variable)
5- Factoring
6- Rational Expressions
7- Quadratic Equations
8- Functions
9- Solving 1st Power Inequalities (1 Variable)
10- Word Problems
11- Extras
st
1

Power Equations (1 Variable)
In order to get the answer, when there is only
one variable

You must, isolate the variable, and if it has a sign
with it (a negative sign) or a number with it, than
you can and must divide the number to the other
side


In order to get the variable completely alone
Then you get your answer
1 Variable Problems

5x + 3 = 2 (2 – 3x)







5x + 3 = 4 – 6x
5x = 4 + (-3) – 6x
5x = 1 – 6x
5x + 6x = 1
11x = 1
X = 1/11



2x = 8

X=4

-x + 20 = – 3x + 2(5x – 10)
 -x + 20 = – 3x + 10x – 20
 -x = 7x – 40
 -8x = -40
 X = 5
These 1 variable problems are
fairly simple and easy
All you have to do is isolate the
variable
Then just add, subtract, or divide
and solve the problem
Algebra Topics
1- Properties
2- Linear Equations
3- Linear Systems
4- Solving 1st Power Equations (1 Variable)
5- Factoring
6- Rational Expressions
7- Quadratic Equations
8- Functions
9- Solving 1st Power Inequalities (1 Variable)
10- Word Problems
11- Extras
Factoring FOIL

FOIL

is a type of factoring that includes two “globs”


(3x + 2)(3x – 2)
this FOIL means that the O and I in FOIL, will be
the same number, but one will be negative and one
positive, therefore, they will cancel each other out
PST

PST, is when two globs are reversed “FOILed”
and they equal perfectly

(x + 2)(x + 2)
2



X + 4x + 4 (PST)
The first number in a PST, to check if you got
a PST, the first number has to be squared, and
if its not, then take out the GCF
The First and Last number should have roots,
while the middle number should be the double
of the roots of both the First and Last number
Factor GCF



The GCF stands for the Greatest Common
Factor
Which means, that if you have a binomial or a
trinomial with prime numbers in common or
more variables than needed, then you can
factor them out, and then continue to solve the
problem
Whatever you factored out, will still be part of
the Answer of the problem
Difference of Squares


First take out the GCF (always)
If there are two globs that if FOILed, arent a PST, but
they just make a binomial, but it can be divided into
two more binomials



Then you have conjugates
(? + ?) (? - ?)
As long as you have a negative glob, that you can still
divide into more globs, you can continue to divide, but if
one glob is the same as another glob, then your answer will
only contain the glob, but only once
Sum or Difference of Cubes

The Sum or Difference of Cubes, is when you
take variable squares or numbers with roots
cubed, and they are separated and into a
binomial and a trinomial
Reverse FOIL


This is the same thing as FOIL factoring, but
there is a Trial and Error system
That means, that when given trinomial, you
will have to guess and check if it FOILs the
correct globs, and you will have to continue to
do that, until you get the correct globs
Factor By Grouping

4x4

It is a binomial because there are two terms, and a
repeated glob, it is a common glob


Which means GCF
2x2

Sometimes you can rearrange the order of the
terms, to find the correct glob
Factor By Grouping

3x1



Rearrange into a PST
Then make two perfect globs
If conjugates then separate them
Algebra Topics
1- Properties
2- Linear Equations
3- Linear Systems
4- Solving 1st Power Equations (1 Variable)
5- Factoring
6- Rational Expressions
7- Quadratic Equations
8- Functions
9- Solving 1st Power Inequalities (1 Variable)
10- Word Problems
11- Extras
Rational Expressions

PST


X 2 + 10x +25

(x + 5)
2
Addition and Subtraction of
Rational Expressions
3
5
+
20
8
=
20
(2)(4)
=
20
2
=
(4)(5)
5
We define a Rational
Expression as a fraction
where the numerator and the
denominator are
polynomials in one or more
variables.
Multiplication and Division of
Rational Expressions
3x2- 4x
2x2- x
x(3x - 4)
=
3x - 4
=
x(2x - 1)
2x - 1
R.E
FOIL
First - multiply the first term in each set of parenthesis: 4x * x = 4x2
Outside - multiply the two terms on the outside: 4x * 2 = 8x
Inside - multiply both of the inside terms: 6 * x = 6x
Last - multiply the last term in each set of parenthesis: 6 x 2 = 12
Websites
Rational Expressions:

http://www.freemathhelp.com/using-foil.html
Algebra Topics
1- Properties
2- Linear Equations
3- Linear Systems
4- Solving 1st Power Equations (1 Variable)
5- Factoring
6- Rational Expressions
7- Quadratic Equations
8- Functions
9- Solving 1st Power Inequalities (1 Variable)
10- Word Problems
11- Extras
Completing the Square (1)




(X + n)² = X² + 2nx + n²
Note the rightmost term (n²) is related
to 2n (the x coefficient) by the formula
Solving this by "completing the square" is as follows:
1) Move the "non X" term to the right:
4X² + 12X = 16




2) Divide the equation by the coefficient of X² which in this case is 4
X² + 3X = 4
3) Now here's the "completing the square" stage in which we:
• take the coefficient of X
• divide it by 2
• square that number
• then add it to both sides of the
equation.
Completing the Square (2)

In our sample problem
the coefficient of X is 3
dividing this by 2 equals 1.5
squaring this number equals (1.5)² = 2.25
Now, adding that to both sides of the equation, we have:


X² + 3X + 2.25 = 4 + 2.25
4) Finally, we can take the square root of both sides of the equation and we have:

X + 1.5 = Square Root (4 + 2.25)

X = Square Root (6.25) -1.5

X = 2.5 -1.5


X = 1.0
Let's not forget that the other square root of 6.25 is -2.5 and so the other root of the
equation is:

(-2.5 -1.5) = -4
Quadratic Formula
We can follow
precisely the same
procedure as
above to derive the
Quadratic
Formula. All
Quadratic
Equations have the
general form:
aX² + bX + c = 0
Discriminant and the Quadratic
Equation





The Discriminant is a number that can be calculated
from any quadratic equation A quadratic equation is
an equation that can be written as
ax ² + bx + c where a ≠ 0
The Discriminant in a quadratic equation is found by
the following formula and the discriminant provides
critical information regarding the nature of the
roots/solutions of any quadratic equation.
discriminant= b² − 4ac
Example of the discriminant
Quadratic equation = y = 3x² + 9x + 5
The discriminant = 9 ² − 4 • 3 •5
Quadratic Equation
Quadratic Equation: y = x² + 2x + 1
•a = 1
•b = 2
•c = 1
•The discriminant for this equation is 2² - 4•1 •1= 4 − 4 = 0
Since the discriminant of zero, there should be 1 real solution to this
equation. Below is a picture representing the graph and one solution of this
quadratic equation Graph of y = x² + 2x +1
Websites
R.E.:



http://www.mathwarehouse.com/quadratic/discriminant-in-quadratic-equation.php
http://webgraphing.com/quadraticequation_quadraticformula.jsp
http://webmath.com/quadtri.html
Algebra Topics
1- Properties
2- Linear Equations
3- Linear Systems
4- Solving 1st Power Equations (1 Variable)
5- Factoring
6- Rational Expressions
7- Quadratic Equations
8- Functions
9- Solving 1st Power Inequalities (1 Variable)
10- Word Problems
11- Extras
F(x)

In Algebra f(x) is another symbol for y


Y=3
F(x) = 3

Its practically the same things, but people use it for
confusion
Domain and Range

Domain


For a function f defined by an expression with
variable x, the implied domain of f is the set of all
real numbers variable x can take such that the
expression defining the function is real. The
domain can also be given explicitly.
Range

The range of f is the set of all values that the
function takes when x takes values in the domain
Domain
Example: The function y = √(x + 4) has the following graph
•The domain of the function is x ≥ −4, since x
cannot take values less than −4. (Try some
values in your calculator, some less than −4 and
some more than −4. The only ones that "work"
and give us an answer are the ones greater
than or equal to −4).
•Note:
•The enclosed (colored-in) circle on the
point (-4, 0). This indicates that the domain
"starts" at this point.
•That x can take any positive value in
this example
Range
Example 1: Let's return to the example
above, y = √(x + 4). We notice that there
are only positive y-values. There is no
value of x that we can find such that we
will get a negative value of y. We say that
the range for this function is y ≥ 0.
Example 2: The curve of y = sin
x shows the range to be
betweeen −1 and 1
The domain of the function y = sin
x is "all values of x", since there are
no restrictions on the values for x.
http://www.intmath.com/Functions-andgraphs/2a_Domain-and-range.php
Algebra Topics
1- Properties
2- Linear Equations
3- Linear Systems
4- Solving 1st Power Equations (1 Variable)
5- Factoring
6- Rational Expressions
7- Quadratic Equations
8- Functions
9- Solving 1st Power Inequalities (1 Variable)
10- Word Problems
11- Extras
Solving Inequalities



Linear inequalities are also called first degree inequalities, as
the highest power of the variable in these inequalities is 1.
E.g. 4x > 20 is an inequality of the first degree, which is often
called a linear inequality.
Many problems can be solved using linear inequalities.
We know that a linear equation with one pronumeral has only
one value for the solution that holds true. For example, the
linear equation 6x = 24 is a true statement only when x = 4.
However, the linear inequality 6x > 24 is satisfied when x > 4.
So, there are many values of x which will satisfy the inequality
6x > 24.
Inequalities

Recall that:

the same number can be subtracted from both sides of an
inequality
the same number can be added to both sides of an inequality
both sides of an inequality can be multiplied (or divided) by
the same positive number
if an inequality is multiplied (or divided) by the same negative



number, then:
Inequalities
Conjunctions

When two inequalities are joined by the word
and or the word or, a compound inequality is
formed. A compound inequality like
-3 < 2x + 5
and 2x + 5 ≤ 7
is called a conjunction, because it uses the
word and. The sentence -3 < 2x + 5 ≤ 7 is an
abbreviation for the preceding conjunction.
Compound inequalities can be solved using the
addition and multiplication principles for
inequalities.
Disjunction

A compound inequality like 2x - 5 ≤ -7 or is
called a disjunction, because it contains the
word or. Unlike some conjunctions, it cannot
be abbreviated; that is, it cannot be written
without the word or.
Algebra Topics
1- Properties
2- Linear Equations
3- Linear Systems
4- Solving 1st Power Equations (1 Variable)
5- Factoring
6- Rational Expressions
7- Quadratic Equations
8- Functions
9- Solving 1st Power Inequalities (1 Variable)
10- Word Problems
11- Extras
Word Problems
1)
The sum of twice a number plus 13 is 75. Find the number.

The word is means equals. The word and means plus.
Therefore, you can rewrite the problem like the following:

The sum of twice a number and 13 equals 75.

Using numbers and a variable that represents something, N in this
case (for number), you can write an equation that means the same
thing as the original problem.
2N + 13 = 75
Solve this equation by isolating the variable.
2N + 13 = 75 Equation. - 13 = -13 Add (-13) to both sides. ------------ 2N = 62



N = 31
Divided both sides by 2
Word Problems
2) Find a number which decreased by 18 is 5
times its opposite.
Again, you look for words that describe equal
quantities. Is means equals, and decreased by
means minus. Also, opposite always means
negative. Keeping that information in mind makes
it so an equation can be written that describes the
problem, just like the following:
N - 18 = 5(-N) Equation. N - 18 = -5N Multiplied out.
5N + 18 5N + 18 Add (5N + 18) to ------------------ both
sides. 6N = 18 N = 3 Divide both sides by 6 to
isolate N.
Word Problems
3) Julie has $50, which is eight dollars more
than twice what John has. How much has
John? First, what will you let x represent?
 The unknown number -- which is how much
that John has.
 What is the equation?
 2x + 8 = 50.
 Here is the solution:
 x = $21
Word Problems
4) Carlotta spent $35 at the market. This was
seven dollars less than three times what she
spent at the bookstore; how much did she
spend there?
 Here is the equation.
 3x − 7 = 35
 Here is the solution:
 x = $14
Algebra Topics
1- Properties
2- Linear Equations
3- Linear Systems
4- Solving 1st Power Equations (1 Variable)
5- Factoring
6- Rational Expressions
7- Quadratic Equations
8- Functions
9- Solving 1st Power Inequalities (1 Variable)
10- Word Problems
11- Extras
11 -
The END