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,
,

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first writ and as or

then divided by

..multiply outer terms of fractions and write a product as a nominator, and a product of inner terms as a denominator

=....simplify

=

=

=

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(,)= (,)
.......eq. 1 and
....eq. 2
__________________________________
.......eq. 1..plug in and
.......solve for





.....plug in and
......solve for

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whoever put it in the vertex form equation is wrong
if you are given the equation
and the vertex at (,)= (,), means that and
then you have , NOT ; here is squared and that is incorrect

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let be anna's and jeff's age
given:
anna is one year younger; means
.............1
the sum of anna's and jeff's age is , means
.......substitute from 1
....solve for



.........anna's age
so, if

........jeff's age
check the product:
the product is

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The ratio of the perimeters is the as the similarity ratio; so, it is .

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a donkey carrying bales of cloth
and
a horse were carrying bales of cloth

The donkey said to the horse "If you give me one of your bales, i shall carry half as much as you."

..............1

If you give me one of yours, replied the horse, I will be carrying 3x's as much as you.
..............2

solve the system:

..............1

..............2
______________________________

..............1..both sides multiply by and solve for
....substitute in 2


..............2..solve for


......the number of bales that donkey was carrying
now find


......the number of bales that horse was carrying

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, , , and

Center: (, )=(, )
Vertices: (-4, -7 ± 9) or
(,), (,)
Co-vertices: (-4 ± 3, -7) or (,), and (,)
Length of Major Axis:
Length of Minor Axis:




2.



, , , and

Center: (, )=(, )
Vertices: (3, -3 ± 4) or
(,), (,)
Co-vertices: (3 ± 3, -3) or (,), and (,)
Length of Major Axis:
Length of Minor Axis:

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if you have

Solved by pluggable solver: Plot a graph of any function AND its DOMAIN

Here's a graph of function (3/5x^2). The domain is shown on the x axis by a green horizontal line.

the domain:
and
the range:


but, if you have

Solved by pluggable solver: Plot a graph of any function AND its DOMAIN

Here's a graph of function ((3/5)x^2). The domain is shown on the x axis by a green horizontal line.

the domain:
and
the range:

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...............1
........2
_________________________________
...............1...solve for
.......substitute in 2
______________________________
........2
.........solve for




...use quadratic formula





solutions:



and




now find

if :




if :




so, solution pairs are:
1. and
2. and

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1.
...factor completely, replace with
.......group


....set


if ..=>...
if ..=>...

so, the roots are and
and points are (,) and (,)
let , solve for and find third point




so, third point is (,)
now, graph it




2.
...factor completely, replace with
.......group


....set


if ..=>...
if ..=>...

so, the roots are and
and points are (,) and (,)
let , solve for and find third point




so, third point is (,)
now, graph it

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4 letter password using letters from A,B,C,D,E,F
By assuming that the first letter is "A", and the last letter is “F",


Therefore, one has to find two more letters from the remaining 4 letters.
Of the 4 letters, there are 1 B, 1 C, 1 D, and 1 E .
The second and third positions can have two different letters.
When the two letters are different. One has to choose two different letters from the 4 available different choices. This can be done in 4 * 3=12 ways.

Total number of possibilities is 12.
1. A,B,C,F
2. A,C,B,F
3. A,B,D,F
4. A,D,B,F
5. A,C,D,F
6. A,D,C,F
7. A,C,E,F
8. A,E,C,F
9. A,D,E,F
10. A,E,D,F
11. A,B,E,F
12. A,E,B,F

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or, you can do it this way:

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..........lambda=λ

where:
= the activity
= the activity at time,


1. How much material was present initially ?

2. How much will be present in 2000 years ?
if




..............

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The term "natural number" refers either to a member of the set of positive integers 1, 2, 3, ... or to the set of nonnegative integers 0, 1, 2, 3, ... e.g., Regrettably, there seems to be no general agreement about whether to include 0 in the set of natural numbers. In fact, Ribenboim (1996) states "Let P be a set of natural numbers; whenever convenient, it may be assumed that 0 in P."
The set of natural numbers (whichever definition is adopted) is denoted N.
Due to lack of standard terminology, the following terms and notations are recommended in preference to "counting number," "natural number," and "whole number."
set................................ name.............. symbol
..., -2, -1, 0, 1, 2, ......... integers.......... Z
1, 2, 3, 4, ............ positive integers.. Z-+
0, 1, 2, 3, 4, .......... non-negative integers... Z-*
0, -1, -2, -3, -4, ......... non-positive integers......

-1, -2, -3, -4, ........... negative integers......... Z--

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The term "natural number" refers either to a member of the set of positive integers 1, 2, 3, ... or to the set of nonnegative integers 0, 1, 2, 3, ... e.g., Regrettably, there seems to be no general agreement about whether to include 0 in the set of natural numbers. In fact, Ribenboim (1996) states "Let P be a set of natural numbers; whenever convenient, it may be assumed that 0 in P."
The set of natural numbers (whichever definition is adopted) is denoted N.
Due to lack of standard terminology, the following terms and notations are recommended in preference to "counting number," "natural number," and "whole number."
set................................ name.............. symbol
..., -2, -1, 0, 1, 2, ......... integers.......... Z
1, 2, 3, 4, ............ positive integers.. Z-+
0, 1, 2, 3, 4, .......... non-negative integers... Z-*
0, -1, -2, -3, -4, ......... non-positive integers......

-1, -2, -3, -4, ........... negative integers......... Z--

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..

as you can see, if two triangles are similar, their sides are proportional; so, the measure of side is proportional to the measure of side

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Vertical Ellipse


.......both sides divide by

, so you have


, , , and

Center: (, )=(, )
Vertices: (, ± ) or
(,), (,)
Co-vertices: ( ± ,) or (,), and (,)
Length of Major Axis:
Length of Minor Axis:

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first convert into :


so

calculate the number of miles in :



now convert into :



plug both values in your formula:

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you will find the and intercept of if you set and solve for , then set and solve for
if




or
.....so, the intercept is at (,)

if




or
.......and the intercept is at (,)

let's check it on a graph

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the statements are:
I) If the cat does not have claws, then the cat cannot scratch the furniture.
II) If the cat can scratch the furniture, then the cat has claws.
III) If the cat has claws, then the cat can scratch the furniture.
you can solve this using truth tables.
you would do so as follows:

p = cat has claws
q = cat scratches furniture
~p->~q = I
q->p = II
p->q = III
truth table:
             p   q   ~p   ~q   ~p->~q   q->p   p->q
             T   T    F    F      T       T     T
             T   F    F    T      T       T     F
             F   T    T    F      F       F     T
             F   F    T    T      T       T     T

from the truth tables, it appears that ~p->~q is equivalent to q->p because their truth tables are the same.
that means that I is equivalent to II
III is not equivalent to either one of them because its truth table is different.
if we examine the statement, we will see why this is so.
the statement are, once again:
I) If the cat does not have claws, then the cat cannot scratch the furniture.
II) If the cat can scratch the furniture, then the cat has claws.
III) If the cat has claws, then the cat can scratch the furniture.
statement II looks like it is the contrapositive of statement I.
statement I says:
if no claws, then no scratch furniture.
statement II says:
if scratch furniture, then claws.
if the statement is:
if no claws, then no scratch furniture
then the contrapositive to that statement is:
if scratch furniture, then claws.
if the statement is:
if scratch furniture, then claws
then the contrapositive to that statement is:
if no claws, then no scratch furniture.
the statement and its contrapositive are equivalent which means if the statement is true then the contrapositive is also true and vice versa.
this is the case between statements I and II.
statement III says:
if claws, then scratch furniture.
the contrapositive to this would be.
if no scratch furniture, then no claws.
all 3 statements are:
I = if no claws, then no scratch furniture
II = if scratch furniture, then claws.
III = if claws, then scratch furniture.
the contrapositive to statement III is:
if no scratch furniture, then no claws
neither I or II are the contrapositive to statement III, therefore the only equivalent statement are I and II.
It's a little convoluted, but that's the way it works out.
Use of the truth tables can make sense of the insensible, assuming you know how to use them.
p->q is only false if p is true and q is false.
q->p is only false if q is true and p is false.
~p->~q is only false if ~p is true and ~q is false.
~q->~p is only false if ~q is true and ~p is false.
that's the truth table for the implied statement.l

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three or more lines in a plane are said to be concurrent if they intersect at a single point
lines divide the plane into regions
check for line
.....it’ true that one line divides the plane into two regions

check for lines
.....it’ true that two lines divides the plane into four regions
check for lines
.....it’ true that two lines divides the plane into six regions

so, now we can check for lines
....so, lines divides the plane into regions

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cups for people: divide by

....simplify, divide both and by

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two sides of a square along with the diagonal of a square form right angle triangle
so, if the of a square garden is , the exact measure of each side could be found using Pythagorean theorem:




.....plug in

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you have a system of linear functions and solution will be coordinates (,) of their intersection point


.....=>......

Solved by pluggable solver: Solve the System of Equations by Graphing

Start with the given system of equations: In order to graph these equations, we need to solve for y for each equation. So let's solve for y on the first equation Start with the given equation Subtract from both sides Rearrange the equation Divide both sides by Break up the fraction Reduce Now lets graph (note: if you need help with graphing, check out this solver) Graph of So let's solve for y on the second equation Start with the given equation Add to both sides Rearrange the equation Divide both sides by Break up the fraction Reduce Now lets add the graph of to our first plot to get: Graph of (red) and (green) From the graph, we can see that the two lines intersect at the point (,) (note: you might have to adjust the window to see the intersection)

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given:
line that passes through (4,0)
you got which is slope-intercept form where slope and y-intercept is
now find an equation in point-intercept form for the line that is parallel to the given line and that passes through the given point

Solved by pluggable solver: Finding the Equation of a Line Parallel or Perpendicular to a Given Line

Since any two parallel lines have the same slope we know the slope of the unknown line is (its from the slope of which is also ). Also since the unknown line goes through (4,0), we can find the equation by plugging in this info into the point-slope formula Point-Slope Formula: where m is the slope and (,) is the given point Plug in , , and Distribute Multiply Add to both sides to isolate y Make into equivalent fractions with equal denominators Combine the fractions Reduce any fractions So the equation of the line that is parallel to and goes through (,) is So here are the graphs of the equations and graph of the given equation (red) and graph of the line (green) that is parallel to the given graph and goes through (,)

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(a) make a sketch of what is happening




(b) When will the rocket be 550 feet from the ground










solution:


.........the rocket be 550 feet from the ground when it goes up
or


........the rocket be 550 feet from the ground when it goes down after it reaches maximum height

(c) After how many seconds will the rocket strike the ground?










solution:



or


.......the rocket strike the ground in ; on a graph, that will be a point (,)