Questions on Algebra: Inverse operations for addition and multiplication, reciprocals answered by real tutors!

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lets call the number x
so x+7(1/x)=16/3---> multiply all terms by 3x
:
3x^2+21=16x
:

x=3 and 2.333(7/3)

Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=4 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 3, 2.33333333333333. Here's your graph:

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so the two negatives will give you a positive,
, then use a common denominator
and
so and 17/35 is the simplest form.

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an excerpt from the web:
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When we say z is jointly proportional to a set of variables, it means that z is directly proportional to each variable taken one at a time.
If z varies jointly with respect to x and y, the equation will be of the form z = kxy (where k is a constant).
Equation: c = 5ab
Variable c is jointly proportional to a and b. That means c is directly proportional to both a and b.
Doubling a causes c to double. Doubling b causes c to double. Doubling both a and b causes c to quadruple.
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end of excerpt.
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based on that definition, the equation as stated would not be joint.
however, if you move the equation around, then that moved around equation could be considered joint.
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your equation p = f/a does not fit the form of a joint equation, but it can be changed to read:
f = a*p
f = a*p looks like it fits the form of a joint variation if you add a k into it (k = 1 in this case), i.e. f = k*a*p.
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why f = a*p would be considered joint, while p = f/a would not is beyond me, but that's what the definition states.
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the proportion has to be direct with each of 2 or more variables taken one at a time, as stated above.
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lets call a the unknown you would write this as follows 5a+8

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The function f(x)=|x+3|+9 is NOT a one-to-one function. The points (19,31) and (__,__) prove that f is not a one-to-one function. What is the missing set of points?
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Solve y = |19+3}+9
y = 31
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Solve 31 = |x+3|+9
22 = |x+3|
x+3 = 22 or x+3 = -22
x = 19 or x = -25
-----------------------
So two of the solution pairs are (19,31) and (-25,31)
This shows that f is not one-to-one.
===================================
Cheers,
Stan H.

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Start with the given equation


Switch 'x' and 'y'. The goal now is to solve for 'y'


Add 5 to both sides.


Take the cube root of both sides to eliminate the exponent of '3'


Rearrange the equation


Subtract 7 from both sides


Divide both sides by 3 to isolate (ie solve for) 'y'


So the inverse function is

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let G=Goeff now

"in 2 years time" __ G+2

"Geoff will be" __ =

"three times as old as he was ten years ago" __ 3(G-10)

G+2=3G-30 __ 32=2G __ 16=G

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In two years time Geoff will be three times as old as he was ten years ago.
a)Using y for his present age, write down an equation connecting his present
age with his age ten years ago?
:
Two things to note in this:
:
Age in two years: (y+2)
Age 10 yrs ago: (y-10)
:
In two yrs = 3 times age 10 yrs ago
y + 2 = 3(y - 10)
:
b)Solve the equation.
:
y + 2 = 3y - 30; multiplied what's inside the bracket
:
y + 2 + 30 = 3y; add 30 to both sides
:
32 = 3y - y; subtract y from both sides
:
y =
:
y = 16 yrs; his age now
:
:
Try this in the original statement, see if works
"In two years time Geoff will be three times as old as he was ten years ago."
16 + 2 = 3(16 - 10)
18 = 3(6)
:
You don't need brackets around the problem

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-b = 8
Do you see the negative sign in front of the letter b?
There is an invisible -1 there.
To find b, divide both sides by -1.
b = 8/-1
b = -8
Did you follow?

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(((d+6(3)=-11)))
d+18=-11
d=-11-18
d=-29 answer.

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When adding and subtracting fractions you need to have the same denominators.
Make the denominators 17*18.






.
.
.
Make the denominators 40.

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The first step is to replace f(x) with y and then switch the y and the x and solve
y=3x-1 isolate for the x variable
add 1 to both sides
y+1=3x
Divide both sides by 3
1/3y+1/3=x This is the inverse function
y=x+3/2x
Simplify this equation
2/2x=1x
2/2+3/2=5/2x
Multiply both sides by 2
2y=5x
Divide both sides by 5
2/5y=x

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First, put your function in terms of x and y.

Now interchange x and y, then solve for y.



That's the inverse of the original function.

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.
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a. _x14=100 a where a is greater than 100. A would be any number greater than 7. to give an answer of more than 100.
b. 4x24=96 b where b is less than 100. B would be any number 4 or less. this would give an answer of less than 100.

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you can translate inches to feet and grams to pounds first, or you can solve the problem and translate inches to feet and grams to pounds later.
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i will solve the problem in inches and grams first and then translate to pounds and feet after i get the result.
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mass is 125 grams in the shape of a cylinder.
volume of cylinder is pi*r^2*h
diameter = 2.48 inches
radius = 1/2 * diameter = 1.24 inches.
height of the cylinder is 1 inch.
let r = radius of the cylinder
let h = height of the cylinder
let pi = the constant 3.141592654
let V = volume of the cylinder
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first find the volume of the cylinder.
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V = pi*r^2*h
height (h) = 1 inch.
radius (r) = 1.24 inches.
formula becomes

V = pi*1.5376*1
V = 4.830512864 cubic inches.
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next find the grams per cubic inch.
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125 grams / 4.830512864 cubic inches = 25.87716947 grams per cubic inch.
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if the problem had asked you for grams per cubic inch you would be done.
unfortunately, the problem asked for pounds per cubic foot.
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actually, the problem asked for pounds per square foot, but this is wrong. the answer has to be in pounds per cubic foot since we are dealing with the volume of a cylinder which is expressed as a cubic measure.
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you divide the number of cubic inches by 1728 to get cubic feet.
(volume of 1 cubic foot is 1 foot * 1 foot * 1 foot which is the same as 12 inches * 12 inches * 12 inches = 1728 cubic inches.
1728 cubic inches is the same as 1 cubic foot.
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since there are 453.592 grams per pound, you divide the number of grams by 453.592 to get the number of pounds.
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125 grams / 453.592 grams per pound = .275578053 pounds.
4.830512864 cubic inches / 1728 = .002795436 cubic feet.
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.275578053 pounds / .002795436 cubic feet = 98.58143186 pounds per cubic foot.
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this becomes 98.58 pounds per cubic foot to the nearest hundredth of a pound per cubic foot.
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i didn't do any rounding and used values stored in the calculator rather than values displayed on the calculator. the answer will be slightly different if you use values displayed on the calculator rather than values stored in the calculator. for example, i did the calculations using the displayed values and the answer was 98.48... rather than the 98.58.... shown above.
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N/2=-15
N=2*-15
N=-30 ANSWER.
PROOF:
-30/2=-15
-30=-30

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It is 1/3/x=x/3 When you multiply two numbers and they =1 they are reciprocals.

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This problem is incorrectly written you cannot add a higher percentage of alloy to a pure metal in order to generate a lower % of alloy. The way this problem is written you would have an alloy composition greater than 65% and less than %100.
I think that the equation was supposed to be using 48% alloy and pure gold to make 65% alloy.
First we would convert the percentages to decimal
100%=100/100=1
48%=48/100=.48
65%=65/100=.65
X=# of grams of pure gold
70-x=# of grams of 48% gold alloy
The equation is as follows
1x+.48(70-x)=.65(70)
Use the distributive property
x+33.6-.48x=45.5
add (1-.48)x=.52x

.52x+33.6=45.5
Subtract 33.6 from both sides
.52x=11.9
Divide both sides by .52
x=22.8 grams of pure gold
Let's check
22.8+[47.11(.48)]=45.5
22.8+22.61=45.5
Both sides =

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The multiplicative identity property.
Cheers,
Stan H.

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opposite numbers: a and -a
opposite reciprocals: 1/a and 1/(-a) = -(1/a)
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Ans: Yes
===============
Cheers,
Stan H.

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Can you be more specific about the number 3 in your equation?
Do we add to 3, or divide by 3 or what?

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Can you be more specific about the number 3 in your equation?
Do we add to 3, or divide by 3 or what?

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Read this --> http://en.wikipedia.org/wiki/Reciprocal_(mathematics)
The multiplicative inverse is just . In this case,
so
=
Now to check, multiply } Do you get 1? If so, you know you did it correctly

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1 is subtracted on the left side, so the inverse operation is adding 1
__ 5x=2x+9

2x is added on the right side, so the inverse operation is subtracting 2x
__ 3x=9

3 is multiplied on the left side, so the inverse operation is dividing by 3
__ x=3



8 is added on the right side, so the inverse operation is subtracting 8
__ 5x-9=2x

5x is added on the left side, so the inverse operation is subtracting 5x
__ -9=-3x

-3 is multiplied on the right side, so the inverse operation is dividing by -3
__ 3=x

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The pattern may seem analogous to a Geomatric Progression Series.
Example can be-:2 + 6+ 12+18_ _ _ _ + upto n

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This problem has been around for a looooong time. It's yet another example of having more unknowns than equations. In problems such as this, it's usually necessary to apply a great deal of trial and error coupled with the constraints inherent in the problem. Many times, there is more than one solution.
Let x=number of chickens
y=number of pigs
z=number of cows
Now we are told that:
x+y+z=100----------------------------eq1
we are also told that:
0.5x+3y+10z=100------------------------eq2
Multiply eq 2 by 2 and then subtract eq1 from it and we get:
5y+19z=100
subtract 19z from each side
5y=100-19z divide each side by 5
y=(100-19z)/5----eq2a --- this equation gives us a relationship between the number of pigs (y) and the number of cows (z).
Now we know a couple of things that will help us to solve this problem:
(1) We cannot have fractions of animals ---we are dealing with whole numbers
(2) We cannot have negative animals----we are dealing with positive numbers
(3) We must, at least, have one of each animal
By inspecting eq2a, we can quickly see that we cannot have more than 5 cows otherwise we start getting negative pigs. So, lets do some trial and error:
If we have 5 cows-----------------y=(100-95)/5=1 BINGO!!
So, 5 cows and 1 pig are possibilities
Lets substitute these values into eq1 and see if they work for chickens:
x+1+5=100
x=94 chickens
Clearly the numbers check; let's see if the values check:
94*0.5+1*3.00+ 5*10=100
47+3+50=100
100=100
So, 94 chicks, 1 pig and 5 cows work
Let's see if we have other possibilities:
If we have 4 cows-----y=(100-76)/5----No!!! we get fractions of pigs
If we have 3 cows-----y=(100-57)/5----No!!! we get fractions of pigs
If we have 2 cows-----y=(100-38)/5----No!!! we get fractions of pigs
If we have 1 cow------y=(100-19)/5----No!!! we get fractions of pigs again
0 cows would also work if the problem allowed}}
Hope this helps---ptaylor

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1/2=.5
The square root of this number= .707106
Therefore, it is IRRATIONAL.

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To begin with:
Rational numbers are basically numbers that DO NOT go on repeating.
For example: 1/2, 3/4, these are numbers/fractions that when divided DO NOT go on repeating.
However, IRrational numbers are numbers that DO go on with repeating decimal.
For example, pi= 3.140596 is an irrational numbers because it DOES GO ON REPEATING.
2/3=.6666666, and because it goes on with a repeating decimal, and the SQUARE ROOT OF THAT NUMBER= .81649...
THEREFORE, IT IS IRRATIONAL.
Hope this helps,
INTEGRAL13

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I'll do the first one to get you going in the right direction

I'm assuming that the notation x->3x means .


Start with the given function


Replace f(x) with y


Switch x and y


Divide both sides by 3 to isolate y


So the answer is which means that the inverse function is

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What you need to do is have x on one side, everything else on the other.
You can use addition/subtraction and multiplication/division to get it done.
Remember, whatever you do to one side of the equation, you must do to the other side.
.
.
.

Subtract 3 from both sides.
Simplify.
Divide both sides by 2.
Simplify.
.
.
.
Always check your answer




True statement, good answer.

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The mileage from Dallas to Miami is 1332 miles. To the nearest hour, how many hours would it take to drive from Dalla to Miami at an average speed of 55mi/h?
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that's 1332/55
= 24 hours. Long trip.

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h/15=10
15h/15=15*10
h=150

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LCM of 14,22,176

Whether you want the LCM or the GCF, you begin by breaking
all the numbers down into a product of prime numbers:

14  = 2*7
22  = 2*11
176 = 2*88 = 2*2*44 = 2*2*2*22 = 2*2*2*2*11

 14 = 2*7
 22 = 2*11
176 = 2*2*2*2*11

To find the GCF, we color red all factors that
are common to ALL THREE

 14 = 2*7
 22 = 2*11
176 = 2*2*2*2*11

There is only one factor in common to all three,
and that is 2.  So
GCF of 14, 22, and 176 is 2 

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To find the LCM, we color red what is in common 
to ANY TWO OR THREE of them:

 14 = 2*7
 22 = 2*11
176 = 2*2*2*2*11

The LCM is to form the product of ONLY ONE of each red factor
but EVERY non-red factor.

LCM = 2*11*2*2*2*2 = 1232 

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LCM,26,14,112

LCM of 26,14,112 

Whether you want the LCM or the GCF, you begin by breaking
all the numbers down into a product of prime numbers:

 26 = 2*13
 14 = 2*7
112 = 2*56 = 2*2*28 = 2*2*2*14 = 2*2*2*2*7

 26 = 2*13
 14 = 2*7
112 = 2*2*2*2*7

To find the GCF, we color red all factors that
are common to ALL THREE

 26 = 2*13
 14 = 2*7
112 = 2*2*2*2*7

There is only one factor in common to all three,
and that is 2.  So
GCF of 26, 14, and 112 is 2 

-----------

To find the LCM, we color red what is in common 
to ANY TWO OR THREE of them:

 26 = 2*13
 14 = 2*7
112 = 2*2*2*2*7

The LCM is to form the product of ONLY ONE of each red factor
but EVERY non-red factor.

LCM = 2*7*13*2*2*2 = 1456 

Edwin

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(-3)(-1/3) = 1
so the inverse of -3 is -1/3
==========================================
Cheers,
Stan H.

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-2.2+1.3=-3.4x-2.3
-.9+2.3=-3.4x
1.4=-3.4x
x=1.4/-3.4
x=-.4118 answer.

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The reciprocal of 6 plus the reciprocal of 8 is the reciprocal of what number?
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1/6 + 1/8 = 1/x
Multiply thru by 24x to get:
4x + 3x = 24
7x = 24
x = 24/7
============
Cheers,
Stan H.

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Find the inverse of the one to one function.
F(x)=4/5x-1
---------------
y = (4/5)x -1
Interchange x and y to get:
x = (4/5)y -1
solve for y to get the inverse:
5x = 4y -5
4y = 5x + 5
y = (5/4)x + 5
That is the inverse.
==========================
Cheers,
Stan H.

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subtracting 1/B and 1/C __ 1/A=D-1/B-1/C

BC is the common denominator __ 1/A=(DBC-C-B)/(BC)

taking the inverse __ A=(BC)/(DBC-C-B)

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if it takes you 6 hours to run four loads with only one washer,
Each load takes 6/4 = 3/2 hr per washer
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how long would it take to run the same at the laudromat using 3 washers?
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(# of loads)/(# of washers)k = (# of hours)
(4/1)k = 6
k = 3/2
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EQUATION:
(# of loads)/(# of washers)(3/2) = # of hours
(4) /( 3) (3/2) = # of hours
4/2 = # of hours
2 = # of hours
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Ans: In 2 hours, 3 washers can do 4 loads.
========================
Cheers,
Stan H.

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If it takes you 6 hours to run four loads with only one washer, how long would it take to run the same at the laundromat using 3 washers?
:
Use washer/hrs to solve this: 1 wash/hr = 1 washer for one hr
:
Let x = time required to do 4 loads in 3 washers
=
or
6 = 3x
x = 2 hrs
:

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In order to solve this problem, one must assume that there are no 1-legged people or 3-legged elephants in the group, i.e. all of the people have 2 legs and therefore 2 knees, and all of the elephants have 4 legs and therefore 4 knees. You also have to assume that there are no elephants which, for whatever reason, have a different number of trunks than 1. Given all of that:

x is the number of people
y is the number of elephants

Then:
2x is the number of knees attributible to people and
4y is the number of knees attributible to elephants.

Since there are 100 knees,

By similar analysis, is the equation that represents the distribution of trunks of both kinds (elephant proboscises and containers for personal belongings).

Your task is then to solve this system of two linear equations in two variables. I recommend the elimination method for this problem.

Hint: Multiply the 2nd equation by -4.

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Yes, unless you count 0 and -0. In which case the reciprocals are undefined.

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Total = 41.50*15
Raised = 400
Total = raised + "whats left"
(41.50*15) - 400 = whatsleft
622.50 - 400 = whatsleft
222.50 = whatsleft
222.50/15 = whatEachPlayerMustPay
14.84

Or looking at another way
Each player costs 41.50
Each player's portion of the 400 = 400/15 = 26.66
41.50-26.66 = 14.84

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What do you mean when you say "need to know x"?
You don't have an equation so you cannot solve for "x".
Cheers,
Stan H.