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Question 279490: The sum of the digits of a two- digit number is 12. If 36 is added to the number, then the number obtained is the original with its digits interchanged. Find the original number.
Found 3 solutions by greenestamps, josgarithmetic, ikleyn:
Answer by greenestamps(13327)   (Show Source):
You can put this solution on YOUR website!

Here is a quick mental solution you can use if formal algebra is not required, and if the speed of finding the solution is important -- as in a timed competitive exam.

The difference between a 2-digit number and the number with the digits reversed is 9 times the difference of the two digits. Since the difference between the two 2-digit numbers is 36, the difference between the two digits is 36/9 = 4.

So the sum of the two digits is 12 and the difference is 4. Quick mental reasoning shows the digits are 4 and 8.

So the original number is 48 and the number with the digits reversed is 84.

ANSWER: 48

Answer by josgarithmetic(39792)   (Show Source):
You can put this solution on YOUR website!
t and u
tens digit and units digit
Find the number, 10t+u.

If add those, then find ; and find from that, .

The requested two digit number,

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
The sum of the digits of a two- digit number is 12. If 36 is added to the number, then the number obtained
is the original with its digits interchanged. Find the original number.
~~~~~~~~~~~~~~~~~~~~~~~~~

        The solution in the post by @mananth, giving the answer '43 ' is incorrect,
        as anybody can check by substituting it to the problem.

        I came to bring a correct and accurate solution.

Let x be in the tens place and y in units place
x+y =12
10x+y+36 = 10y+x
10x-10y+y-x+36=0
9x-9y=-36
x+y=12
9x+9y=108
18x=72
x=4
x+y=12 but x=4 so y=8.

The number is 48.         ANSWER

Solved correctly.

Question 51100: Hi I would really appreciate your help
Please help me solve -.5x - 5 > 11
I don't understand how to solve this with .5 of an x
Please Help
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39792)   (Show Source):
You can put this solution on YOUR website!

means exactly the same as

Answer by MathTherapy(10806)   (Show Source):
You can put this solution on YOUR website!

Hi I would really appreciate your help Please help me solve -.5x - 5 > 11 I don't understand how to solve this with .5 of an x Please Help ************************************* -.5x - 5 > 11 - .5x > 11 + 5 - .5x > 16 -- Dividing by a negative value changes the inequality from > to <

Question 1007654: A person rowed their boat downstream for 100 miles and they took 2 hours. Returning upstream, the trip took 2 hours and 40 minutes.
What is the speed of the water?

Found 2 solutions by n2, ikleyn:
Answer by n2(79)   (Show Source):
You can put this solution on YOUR website!
.
A person rowed their boat downstream for 100 miles and they took 2 hours.
Returning upstream, the trip took 2 hours and 40 minutes.
What is the speed of the water?
~~~~~~~~~~~~~~~~~~~

Let x be the rate of the boat in still water (in miles per hour) and y be the rate of the current (in the same units). Then the effective rate of the boat downstream is x + y and the effective rate of the boat upstream is x - y. From the problem, the effective rate of the boat downstream is the distance of 100 miles divided by the time of 2 hours = 50 mph. The effective rate of the boat upstream is the distance of 100 miles divided by the time of 2 hours and 40 minute, or 2 hours, or hours = = 37.5 mph So, we have two equations for 'x' and 'y' x + y = 50, (1) x - y = 37.5. (2) To find 'y', subtract equations (2) from equation (1). The terms 'x' and 'x' will cancel each other, and you will get 2y = 50 - 37.5 = 12.5 ---> y = 12.5/2 = 6.25. At this point, the solution is complete. ANSWER. The rate of the current is 6.25 miles per hour.

Solved.

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
A person rowed their boat downstream for 100 miles and they took 2 hours.
Returning upstream, the trip took 2 hours and 40 minutes.
What is the speed of the water?
~~~~~~~~~~~~~~~~~~~~~~~~~~

        In the post by @mananth, the solution is produced by a computer code.

        Neither the style of the solution, nor its form of presentation are perfect;
        they are difficult to read and to understand.

        So, I present here my solution in simple, straightforward and clear, transparent form,
        as it should be done to every school Math problem.

Let x be the rate of the boat in still water (in miles per hour) and y be the rate of the current (in the same units). Then the effective rate of the boat downstream is x + y and the effective rate of the boat upstream is x - y. From the problem, the effective rate of the boat downstream is the distance of 100 miles divided by the time of 2 hours = 50 mph. The effective rate of the boat upstream is the distance of 100 miles divided by the time of 2 hours and 40 minute, or 2 hours, or hours = = 37.5 mph So, we have two equations for 'x' and 'y' x + y = 50, (1) x - y = 37.5. (2) To find 'y', subtract equations (2) from equation (1). The terms 'x' and 'x' will cancel each other, and you will get 2y = 50 - 37.5 = 12.5 ---> y = 12.5/2 = 6.25. At this point, the solution is complete. ANSWER. The rate of the current is 6.25 miles per hour.

Solved.

----------------------

For a computer code (which @mananth uses to create his solution files),
there is no difference which style of the solution to produce.

But for a human reader, there is a huge difference what to read and from which source to learn.

So, I created my solution here in order for you see the difference.

Question 1108030:
A baseball team has home games on Friday

and Sunday
.
The two games together earn $3268.50

for the team. Friday
's
game generates $603.50

less than Sunday
's
game. How much money was taken in at each game?
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
A baseball team has home games on Friday and Sunday
The two games together earn $3268.50 for the team.
Friday's game generates $603.50 less than Sunday's game.
How much money was taken in at each game?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        The solution in the post by @mananth contains several arithmetic errors that lead to wrong answer.
        So, I came to bring a correct accurate solution.

Let Sunday's game generate $ x Friday's game generates x - 603.50 dollars. The two games generate 3268.50 x + (x-603.50) = 3268.50 2x - 603.5= 3268.50 2x = 3268.50 + 603.50 2x = 3872 x = 3872/2 x = 1936 ANSWER. Sunday's game generated $1936. Friday's game generated $1936 - $603.50 = $1332.50.

Solved correctly.

Question 597322: Please help me solve this equation
Answer by MathTherapy(10806)   (Show Source):
You can put this solution on YOUR website!

Please help me solve this equation ***************************************************************** There ISN'T just 1 value for x, as the other person indicates! ---- Squaring each side ------ Squaring each side 4x = 0 OR x - 4 = 0 x = 0 OR x = 4 Both solutions are VALID!!

Question 1180164: -x + 3y = 2 and x - 5y = -4 Solving for Two Variables using Elimination
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
-x + 3y = 2 and x - 5y = -4 Solving for Two Variables using Elimination
~~~~~~~~~~~~~~~~~~~~~~~~~

        I write it, because I do not agree with the procedure, which @mananth uses in his solution.
        As the equations are given, there is a straightforward way to proceed, without making unnecessary calculations.
        You will get the same answer, but will not do unnecessary job.

-x + 3y = 2, (1) x - 5y = -4. (2) As equations are given, you are lucky, because equations ARE JUST READY for elimination. Add equations (1) and (2). The terms '-x' and 'x' will cancel each other, so you will get 3y + (-5y) = 2 + (-4), -2y = -2, y = (-2)/(-2) = 1. Then from equation (2) x = -4 + 5y = -4 + 5*1 = -4 + 5 = 1. ANSWER. The solution is x = 1, y = 1.

Solved in a way as it should be done and as it is expected to be done.

If you will solve following to @mananth's way, the teacher will make squared eyes
and will consider your job as if you are mentally blind and do not see the simplest way.
In the next instance, the teacher will understand, that you were indoctrinated
by an Artificial Intelligence in its, still not perfect, mode.

//////////////////////////

I do understand PERFECTLY, why @mananth selected that way.

It is because his computer code, which he uses as an Artificial Intelligence,
is programmed this way.

It does not seek for the best way or for most reasonable way.

In this sense, his computer code is not a perfect tool for teaching.

Question 44251: can someone please help me with this problem: 1/3x+6 +2 = 3/x+2
Answer by MathTherapy(10806)   (Show Source):
You can put this solution on YOUR website!

can someone please help me with this problem: 1/3x+6 +2 = 3/x+2 The answer provided by the other person who responded, is WRONG!! Based on the person's statement, if he/she wants to do well in the tutoring business, the person needs to know how to check his/her work, before submitting it. And, after doing so, this person also needs to CHECK whatever answer he/she has derived. 1 + 2(3)(x + 2) = 3(3) --- Multiplying by LCD, 3(x + 2) 1 + 6x + 12 = 9 6x = 9 - 13 6x = - 4 If you DON’T use an LCD, but a COMMON DENOMINATOR instead, you could end up with a quadratic equation that can be FACTORIZED nicely, and result in another solution, - 2. However, this solution is an EXTRANEOUS one, as it leads to both original fractional-expressions, being UNDEFINED!

Question 1210515: in a certain freshman class the number of girls is 83 less than twice the number of boys(b) the total number of students in that freshman class is 259 how many girls and boys are in that class
Answer by josgarithmetic(39792)   (Show Source):
You can put this solution on YOUR website!

Question 456599: How do I substitute a problem like this?
y = x + 3
y = 2x + 12
Found 2 solutions by timofer, ikleyn:
Answer by timofer(155)   (Show Source):
You can put this solution on YOUR website!
The two expressions for y are equal. How do you substitute? Simple!

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
How do I substitute a problem like this?
y = x + 3
y = 2x + 12
~~~~~~~~~~~~~~~~~~~~~~~~

In his post, @mananth obtained the solution x = -9, y = -6.

This is correct.

But then, @mananth points that the ordered solution is (-6,-9).

This is NOT an ordinary/commonly accepted order.

The ordinary/commonly accepted order is opposite (x,y) = (-9,-6).

Question 454327: Solve the linear system by using substitution. 3x+2y=5, 5x-9y=-4
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39792)   (Show Source):
You can put this solution on YOUR website!
The first one has a 3x, and the second one has a 5x.

Same system

Substituting for 15x in the second equation

Now use this any way you want to find x.

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
Solve the linear system by using substitution. 3x+2y=5, 5x-9y=-4
~~~~~~~~~~~~~~~~~~~~~~~~

        In his post, @mananth makes 16 tons incorrect calculations.
        A true solution, which I provide below, has nothing in common with the @mananth writing.

Your starting equations are 3x + 2y = 5, (1) 5x - 9y = -4. (2) From equation (1), express y = and substitute it into equation (2), replacing 'y' there 5x - = -4. Multiply both sides by 2 to rid of the numerator 10x - 9*((-3x)+5) = -8. Simplify and find 'x' from this equation 10x + 27x - 45 = -8 37x = -8 + 45 37x = 37 x = 37/37 = 1. Now substitute x = 1 into equation (1) and find 'y' 3*1 + 2y = 5 ---> 2y = 5 - 3 = 2 ---> y = 2/2 = 1. ANSWER. x = 1, y = 1.

Solved.

Question 451898: How do I set this problem up:
If a farmer stand owner mixes apple juice and cranberry juice, how much should he charge if he mixes 8L of apple juice selling for 45 cents a liter with 10L of cranberrt juice sellling for $1.08 a liter.
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
How do I set this problem up:
If a farmer stand owner mixes apple juice and cranberry juice, how much should he charge
if he mixes 8L of apple juice selling for 45 cents a liter with 10L of cranberry juice selling for $1.08 a liter.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        In the post by @mananth, the solution is FATALLY WRONG and explained INCORRECTLY.
        I came to make the job in a right way as it SHOULD be done.

The total price for the mixture is 0.45*8 + 1.08*10 = 14.40 dollars. The total volume of the mixture is 8 liters + 10 liters = 18 liters. Therefore, it is logical to assume that this mixture should be sell at the price = 0.8 dollars per liter, or 80 cents per liter. In compact form, the formula for the price per liter is = 0.80 dollars per liter. ANSWER

Solved and explained in a right way, as it should be done.

For the peace in your mind, ignore that nonsense in the post by @mananth.

Question 449484: Solve by the elimination method.
0.3x-0.2y=4
0.2x+0.5y=-75/17
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
Solve by the elimination method.
0.3x-0.2y=4
0.2x+0.5y=-75/17
~~~~~~~~~~~~~~~~~~~~~~~~~~

The solution x = -0.77, y = -8.52 in the post by @mananth is INCORRECT.

I checked it by substituting his values into the first equation ang got the value 1.473,
different from 4, which is in the right side of this equation.

It is just enough to refute the @mananth solution.

Question 447774: how to simplify sqrt (12ab^3c^2)
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
how to simplify sqrt (12ab^3c^2)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        The answer      in the post by @mananth is .

        The correct answer is   .

                                E x p l a n a t i o n

In problems of this kind, we assume that the variables are real numbers and the expression under the square root is non-negative. We should understand that c^2 is necessary (always) non-negative, but 'c' itself can be negative. We also should understand that ' b ' can be negative (together with b^3), but the product ' a*b^3 ' is necessary non-negative, same as the product ' ab '. In the problems of this kind, it is also assumed that the value of the square root itself is also non-negative. Therefore, when ' b^2 ' goes out the square root, it comes to outside as |b|, the absolute value of ' b '. Similarly, when ' c^2 ' goes out the square root, it comes to outside as |c|, the absolute value of ' c '. Therefore, the final formula in the answer is as it is shown in my post at the beginning.

Solved correctly and explained completely.

Question 445209: i'm trying to show a=b or a=1/b without plugging in the values for a.
this is the equation
a+1/a=b+1/b
i have tried multiplying both sides by ab
i tried the quadratic equation but that didn't work very well cause i couldn't get the square root out
i just need a step by step explanation of how to get there cause its driving me crazy and i think i'm just missing something simple
please remember. i know how to substitute. i'm basically proving a= b,1/b
which is why i tried making it a quadratic because i knew there were 2 answers
thank you very much
i hope you can help
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
i'm trying to show a=b or a=1/b without plugging in the values for a.
this is the equation
a+1/a=b+1/b
i have tried multiplying both sides by ab
i tried the quadratic equation but that didn't work very well cause i couldn't get the square root out
i just need a step by step explanation of how to get there cause its driving me crazy and i think i'm just missing something simple
please remember. i know how to substitute. i'm basically proving a= b,1/b
which is why i tried making it a quadratic because i knew there were 2 answers
thank you very much
i hope you can help
~~~~~~~~~~~~~~~~~~~~~~~~~

Good question and good problem.

@mananth was on a right track, but his solution was incomplete.
I came to present a complete solution/answer/exposition/explanation.

a + = b + = multiply by ab = = = a - b ab(a-b)=(a-b) ab(a-b) - (a-b) = 0 (ab-1)*(a-b) = 0 So, EITHER ab = 1, which is the same as a = (one of the two your options) OR a = b (second of the two your options).

Solved completely.

My congrats !

Question 444146: 6x - y = 39
6x + 7y=33
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39792)   (Show Source):
You can put this solution on YOUR website!

, and

Now use this to find value for x.

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
6x - y = 39
6x + 7y=33
~~~~~~~~~~~~~~~~~~~

As I look at the solution in the post by @matanth, his solution evokes a feeling of protest in me,
because, strictly speaking, it is incorrect.
Using decimal numbers with two digits after the decimal dot, he found the solution x = 6.38, y = -0.75.

While the 'y'-value is precise, 'x'-value is approximate.

The exact 'x'-value is 6.375, so the exact solution is (x,y) = (6.375,-0.75).

As I see, the computer code, which @manath uses for his calculations, is very restricted
and does not make a selection, which form to use - in rational numbers or in decimals,
and always prints the output decimals with 2 decimal digits.

In many cases, it leads to incorrect/erroneous solution.

A good code (I mean, a truly good code) must be much smarter and much more flexible,
with built-in-brain.

Question 436132: Word problem.
It takes 60 minutes to fill a pond with water alone.
It takes 80 minutes to drain a pond of water alone.
How long will it take to fill the pond while it is being drained at the same time?
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39792)   (Show Source):
You can put this solution on YOUR website!

rate time amount filling 1/60 60 1 draining 1/80 80 1 combined 1/60-1/80 x 1

The rates are additive, but need to understand draining is negative while filling is positive.
ponds per minute

That is 1 pond in 240 minutes, or 240 minutes to fill the 1 pond with both flows opened.

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
Word problem.
It takes 60 minutes to fill a pond with water alone.
It takes 80 minutes to drain a pond of water alone.
How long will it take to fill the pond while it is being drained at the same time?
~~~~~~~~~~~~~~~~~

        The solution in the post by @mananth is FATALLY WRONG due to elementary arithmetic error.
        I came to bring a correct solution.

consider filling or emptying the pool as 1 job.
emptying the pool = 80 minutes
so 1/80 of the job is done in 1 minute
...
Filling time = 60 minutes
1/60 of the job in 1 minute
..
difference = 1/60 -1/80
(80-60)/80*60
20/4800
1/240 of the job is done in 1 minute when both are on.
so it will take 240 minutes = 4 hours to fill the pool.

Solved correctly.

///////////////////////////////////

It can be seen even with unarmed eye, using common sense - ONLY.

The filling hose fills the tank in 1 hour - but then the drain takes out 3/4 of the water
from the tank during this 1 hour,
so only 1/4 of the tank is filled in one hour.

Hence, 4 hours are needed.

Question 421830: -4x-2y=-8 y=-2x+4
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
-4x-2y=-8
y=-2x+4
~~~~~~~~~~~~~~~~~~~~~

        The conclusion and the answer in the post by @mananth both are incorrect.

        His answer is " There is no solution to this system ".

        The correct answer is OPPOSITE :   there are infinitely many solutions to this system.

Indeed, the first equation can be equivalently transformed/re-written -4x - 2y = -8, 4x + 2y = 8, 2x + y = 4, y = -2x + 4. Then you see that this transformed equation is IDENTICAL to the second equation. Thus the system consists of two equivalent equation - such systems have infinitely many solutions. You can take any value for 'x' and calculate the relevant value of 'y'. Then the pair (x,y) will be solution for both equations, and doing this way, you get infinitely many solutions.

Again, the answer in the post by @mananth is INCORRECT.
The correct answer is OPPOSITE.

Question 730274: A company manufactures an alarm clock. Three weeks ago the had 250 on hand. Two weeks from now it will have 500. Assume the company will continue to make the clocks at this same rate

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
A company manufactures an alarm clock. Three weeks ago the had 250 on hand. Two weeks from now it will have 500.
Assume the company will continue to make the clocks at this same rate
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Very interesting and intriguing, but incomplete.

Question 733492: how would i solve this equation : (2t)^4*3^3
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39792)   (Show Source):
You can put this solution on YOUR website!

You mean, simplify. Not solve equation.

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
how would i solve this equation : (2t)^4*3^3
~~~~~~~~~~~~~~~~~~~~~~

There is NO equation in your post.

What you mistakenly call as " equation ", in reality is an " expression ".

So, an expression is the correct term for your object in Math.

Regarding expressions, the term " solve " is never used.

Instead, the term " simplify " is used.

Or, alternatively, the term " evaluate ".

But not " solve ".

Question 737462: The width of a rectangle is 18 feet less than the perimeter. the area of the rectangle is 2,040 square feet. What are the dimensions of the rectangle?
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
The width of a rectangle is 18 feet less than the perimeter. the area of the rectangle is 2,040 square feet.
What are the dimensions of the rectangle?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

W = (2W + 2L) - 18 Hence W = 18 - 2L. Write an equation for the area of the rectangle LW = 2040 square feet L*(18-2L) = 2040 18L - 2L^2 = 2040 2L^2 - 18L + 2040 = 0 L^2 - 9L + 1020 = 0 Look at the discriminant d = b^2 - 4ac = (-9)^2 - 4*1*1020 = 81 - 4080 is negative number. It tells that this quadratic equation has no real solutions. Hence, the problem is defective and describes a situation which never may happen in real world.

Solved, with explanations.

Question 612274: I have two algebra I word problems i need help setting up. I would like if you can explain the steps of setting up.
Here it is:
Problem #1
The length of a rectangle is 2 1/2 times its width. Its area is 90 square units. What are its dimensions? (Hint: length times width = area)
Let w= width in units.
L=(2 1/2)w= 5w/2 = length in units

Answer by MathTherapy(10806)   (Show Source):
You can put this solution on YOUR website!

I have two algebra I word problems i need help setting up. I would like if you can explain the steps of setting up. Here it is: Problem #1 The length of a rectangle is 2 1/2 times its width. Its area is 90 square units. What are its dimensions? (Hint: length times width = area) Let w= width in units. L=(2 1/2)w= 5w/2 = length in units The person who responded is WRONG!! Width is NOT 3 units! As indicated by you, let width be w Then length (L), as you stated is , or 2.5w As it's given that area = 90 square units, we get: 2.5w(w) = 90 2.5w² = 90 Width of rectangle, or Length: 2.5(6) = 15 units

Question 747745: Some of Aaron's friends are planning to buy him a gift worth 270, dividing the cost equally among themselves. Six more of his friends decided to share in the expenses and so each one's share is decreased by 12. How many friends were originally part of the plan?
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
Some of Aaron's friends are planning to buy him a gift worth 270, dividing the cost equally among themselves.
Six more of his friends decided to share in the expenses and so each one's share is decreased by 12.
How many friends were originally part of the plan?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Let 'n' be the number of friends originally. Then each of n friends contributes . In the extended version, the number of friends is (n+6), and every contributes . Write equation as you read the problem - = 12. For simplicity, divide both sides by 3 - = 4. Multiply both sides by n*(n+6) 90(n+6) - 90n = 4n*(n+6), 90n + 540 - 90n = 4n*(n+6), 540 = 4n*(n+6). Divide both sides by 4 135 = n*(n+6), n^2 + 6n - 135 = 0. Factor (n-9)*(n+15) = 0. Thus the roots of the quadratic equation are 9 and - 15, and we select positive n= 9 as the solution to the problem. ANSWER. The original number of friends was 9.

Solved.

Question 869743: The sum of 4 times a number and 2 is 18. What is the number?
Answer by timofer(155)   (Show Source):
You can put this solution on YOUR website!
sum of 4 multiplied by a number
4x+

... and 2, is 18.

Solve-steps

Question 479860: I am working on my math packet for school. I came across a section in it, in which i completely forgot how to do. i asked my family and they helped me on most of the problems but they couldn't help me figure this one out.
I am suppose to determine the answer for each problem. Simplify when possible:
Here is the equation:
I've tried combining the "x"s and I have also tried using the order of operations but no matter what i use, it seems like i can't solve it.
If you could go through each step and explain what you did and why you did it, i would make it much easier for me to understand and i would greatly appreciate it.

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
I am working on my math packet for school. I came across a section in it, in which i completely forgot
how to do. i asked my family and they helped me on most of the problems but they couldn't help me figure this one out.
I am suppose to determine the answer for each problem. Simplify when possible:
Here is the equation:
I've tried combining the "x"s and I have also tried using the order of operations but no matter what i use, it seems like i can't solve it.
If you could go through each step and explain what you did and why you did it, i
would make it much easier for me to understand and i would greatly appreciate it.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        How tutor @Theo solves the problem and explains the solution to you, this is absolutely irrelevant.

        @Theo tries to solve an equation, which he invented on your own - the problem does not
        requires you to solve any equation.

        It only asks you to simplify this expression   3(2x-1) - (x-1).

To do it, you open parentheses according to the distributive rule, apply the rule of signs, group the like terms and then combine like terms. Everything of it can be done in one line 3*(2x-1) - (x-1) = 6x - 3 - x + 1 = (6x-x) -3 + 1 = 5x - 2. <<<---=== ANSWER

Solved.

It is SO SIMPLE, that you rarely can find simpler assignment in Math.

You only need to know the basic rules of working with polynomial expressions.

-------------------------------

What you are given and what you are intended to transform, is not " an equation ".

It is " an expression ", instead.

Many struggles of students begin from the fact that they do not know
and do not understand the difference between these two basic conceptions.

Question 500890: please help me with this question.
Express in terms of odd and even numbers why the number 286 would not appear in the series 4, 12, 24, 40... ?

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.

The question in this post is posed incorrectly.

The correct question should ask

Express in terms of divisibility of integer numbers why the number 286 would not appear in the series 4, 12, 24, 40... ?

No one professionally written textbook will present an assignment in this form.

It tells me that you try to compose a Math problem on your own,
without having knowledge on how it should be done in Math.

Question 553434: solve the system of equations using the substitution method
x+z=8
y-z=5
x-y=9
Found 2 solutions by MathTherapy, ikleyn:
Answer by MathTherapy(10806)   (Show Source):
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solve the system of equations using the substitution method x+z=8 y-z=5 x-y=9 x + z = 8____z = 8 - x ---- eq (i) y - z = 5 ----- eq (ii) x - y = 9___x - 9 = y ----- wq (iii) y - z = 5 x - 9 - (8 - x) = 5 ------ Substituting x - 9 for y, and 8 - x for z in eq (ii) x - 9 - 8 + x = 5 x + x - 17 = 5 2x = 22 x - 9 = y 11 - 9 = y ------ Substituting 11 for x in eq (iii) z = 8 - x z = 8 - 11 ------ Substituting 11 for x in eq (i)

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
solve the system of equations using the substitution method
x+z=8
y-z=5
x-y=9
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        Yes, the substitution method is robust and works perfectly, as tutor @Theo showed in his post.
        But I will show you even more effective method.

Add all the three equations. Then, after combining like terms, you will get 2x = 8 + 5 + 9 = 22. Hence, x = 22/2 = 11. Having it, you find from the first equation z = 8 - x = 8 - 11 = -3, and from the second equation y = 5 + z = 5 + (-3) = 2. ANSWER. x = 11, y = 2, z = -3.

Solved.

It is a kind of the Elimination method.

Resume:   the Substitution method is good, but sometimes other methods
                are more effective than the Substitution method.

Question 1160143: How would I write the following so the coefficient is in front?
1. (3y)8
2. 3(4b)
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
How would I write the following so the coefficient is in front?
1. (3y)8
2. 3(4b)
~~~~~~~~~~~~~~~~~~~~~~~~

The answer (1) in the post by @MowMow is INCORRECT.

The correct answer to (1) is (3*8)*y = 24y.

Question 1209923: Let x, y, and z be real numbers. If x^2 + y^2 + z^2 = 1, then find the maximum value of
3x + 4y + 5z + x^3 + \frac{4x^2*y)/{z} + \frac{z^5}{xy^2}
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
Let x, y, and z be real numbers. If x^2 + y^2 + z^2 = 1, then find the maximum value of
3x + 4y + 5z + x^3 + + .
~~~~~~~~~~~~~~~~~~~~~~~~~~~

            @CPhill finds it difficult to give a definitive answer in his post.

Meanwhile, the answer to this question is very simple: under given conditions,
the given function/expression has NO maximum.

It is because the term of the expression has variable z in the denominator.

        Take (x,y,z) in vicinity of (,,),   so that   x^2 + y^2 + z^2 = 1   is valid,

                  and let z goes to zero from the positive side.

Then the term tends to positive infinity,
and with this term, the whole expression tends to positive infinity.

////////////////////////////////

So the answer in the post by @CPhill is empty.

Question 1210272: (-9x + 57)^x = 729,
find x.
Answer by Edwin McCravy(20077)   (Show Source):
You can put this solution on YOUR website!

When the unknown in an equation appears as an exponent as well as a term or factor, there is no way to solve it but by trial and error or some iterative method which amounts to trial and error. Since 729 = 3⁶, you might try 6: So 6 is a solution. With a graphing calculator, you can also get another solution, although this one is approximate: x = 1.7746996 Edwin

Question 1209973: Let a and b be positive real numbers. Let
m = \min \left\{ a, \frac{1}{b}, b^2 + \frac{1}{a + 1} \right\}.
Find the largest possible value of m.

Found 4 solutions by mccravyedwin, ikleyn, Edwin McCravy, CPhill:
Answer by mccravyedwin(421)   (Show Source):
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I didn't bother to solve the system of equations, but let online technology do that for me. Thanks to Ikleyn for solving them. But the question that is still not answered is: WHY does the maximum of the minimums occur when the three expressions are equal? It seemed to me like that would be the case, but WHY? Edwin

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
Let a and b be positive real numbers.
Let m = min { a, 1/b, b^2 + 1/(a + 1).
Find the largest possible value of m.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

We want to have a = , (1) + = (2) Substitute expression (1) into equation (2). You will get + = . Simplify this three-story fraction in the left side + = . Multiply both sides by b*(b+1). You will get b^3*(b+1) + b^2 = b+1. Write in the canonical polynomial form b^4 + b^3 + b^2 - b - 1 = 0. It can not be solved algebraically. Solve it numerically using specialized software www.desmos.com/calculator. The only positive root is b = 0.84837 (approximately). See the plot, showing the root, under this link https://www.desmos.com/calculator/tett0qlsmc <<<---=== it is the report on numerical solution (Click on x-interception point to see the value of the root). So, the solutions are b= 0.84837, a = = = 1.17873 (close to the values in the post by Edwin). The value of the expression + is + = 1.17871. Thus the approximate solution, derived this way, is a = 1.17873, b = 0.84837, 1/b = 1/17873 = a, + = 1.17871. This set of values provides the maximum possible value of m = min ( a, , ) over all pairs of positive real numbers {(a,b)}.

Solved.

Answer by Edwin McCravy(20077)   (Show Source):
You can put this solution on YOUR website!

I don't believe AI has the right answer. Let a and b be positive real numbers. Let Find the largest possible value of m. I can't prove it right off, but my better judgment tells me it's whatever values of ' a ' and ' b ' that make all three expressions equal, IF there exist any such values. I'll work on proving it, and if I succeed I'll post it. I hold an MS degree in mathematics from Clemson University, but I never studied any such topics as this in any course I ever took. WolframAlpha tells me such values do exist, and they are approximately a=1.1787241761052217926, b=0.848374895731953217 They check with my TI-84 calculator. So I believe the maximum is approximately 1.1787241761052217926 Edwin

Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website!
Let $m = \min \left\{ a, \frac{1}{b}, b^2 + \frac{1}{a+1} \right\}$.
We want to find the largest possible value of $m$.
Since $m$ is the minimum of the three quantities, we have
$$ m \le a, \quad m \le \frac{1}{b}, \quad m \le b^2 + \frac{1}{a+1}. $$
We want to maximize $m$, so we consider the case when all three quantities are equal to $m$:
$$ a = m, \quad \frac{1}{b} = m, \quad b^2 + \frac{1}{a+1} = m. $$
From $a=m$ and $\frac{1}{b} = m$, we get $a=m$ and $b=\frac{1}{m}$.
Substitute $a=m$ and $b=\frac{1}{m}$ into the third equation:
$$ \left( \frac{1}{m} \right)^2 + \frac{1}{m+1} = m. $$
$$ \frac{1}{m^2} + \frac{1}{m+1} = m. $$
Multiplying by $m^2(m+1)$, we get
$$ m+1 + m^2 = m^3(m+1). $$
$$ m+1 + m^2 = m^4 + m^3. $$
$$ m^4 + m^3 - m^2 - m - 1 = 0. $$
Let $f(m) = m^4 + m^3 - m^2 - m - 1$. We are looking for a positive real root.
We can test some values:
$f(1) = 1+1-1-1-1 = -1$
$f(2) = 16+8-4-2-1 = 17$
Since $f(1) < 0$ and $f(2) > 0$, there is a root between 1 and 2.
Let's consider the case when the three quantities are close to each other.
If we set $a = \frac{1}{b}$, then $ab = 1$.
Then we have $a = m$, $b = \frac{1}{m}$, and $b^2 + \frac{1}{a+1} = m$.
$$ \frac{1}{m^2} + \frac{1}{m+1} = m. $$
Let us try to find a simpler solution.
Suppose $a = \frac{1}{b} = b^2 + \frac{1}{a+1} = m$.
If $a = \frac{1}{b}$, then $ab=1$.
If $a = m$ and $\frac{1}{b} = m$, then $b=\frac{1}{m}$.
If $m = b^2 + \frac{1}{a+1}$, then $m = \frac{1}{m^2} + \frac{1}{m+1}$.
If we assume $m = 1$, then $a = 1$, $b = 1$, and $1 = 1^2 + \frac{1}{1+1} = 1 + \frac{1}{2} = \frac{3}{2}$, which is false.
Let's try to find a solution by setting $a = \frac{1}{b}$.
Then $m \le a$ and $m \le \frac{1}{b}$, so $m \le a$ and $m \le a$.
Also, $m \le b^2 + \frac{1}{a+1}$.
Since $b = \frac{1}{a}$, $m \le \frac{1}{a^2} + \frac{1}{a+1}$.
If $m = a$, then $a \le \frac{1}{a^2} + \frac{1}{a+1}$.
$a^3(a+1) \le a+1 + a^2$
$a^4 + a^3 \le a^2 + a + 1$
Let's consider the case when $a = \frac{1}{b} = b^2 + \frac{1}{a+1}$.
If $a=1$, then $b=1$.
$1 = 1^2 + \frac{1}{1+1} = 1 + \frac{1}{2} = \frac{3}{2}$, which is false.
If $m=1$, then $a=1$ and $b=1$.
Then $m = \min \left\{ 1, 1, 1^2 + \frac{1}{1+1} \right\} = \min \left\{ 1, 1, \frac{3}{2} \right\} = 1$.
Thus, $m=1$ is a possible value.
Let $a = 1.32$. $b = 1/a = 0.7575$. $b^2 + 1/(a+1) = 0.7575^2 + 1/2.32 = 0.5738 + 0.4310 = 1.0048$
If $m=1$, then $1 \le a$, $1 \le 1/b$, $1 \le b^2 + 1/(a+1)$.
$b \le 1$, $b^2 \ge 1-1/(a+1)$
$b^2 \ge a/(a+1)$.
$1/b \ge 1$, so $1/b \ge a$.
$1 \ge a/(a+1)b^2$, so $1/b^2 \ge a/(a+1)$.
$1/b^2 = a^2 \ge a/(a+1)$.
$a^2(a+1) \ge a$.
$a(a+1) \ge 1$.
$a^2+a-1 \ge 0$.
$a = (-1 \pm \sqrt{1+4})/2 = (-1 \pm \sqrt{5})/2$.
$a \ge (-1 + \sqrt{5})/2$.
If $m=1$, then $a \ge 1, b \le 1$, and $b^2 + \frac{1}{a+1} \ge 1$.
If $a=1$, $b=1$, $1 \le 1, 1 \le 1, 1 \le 3/2$.
Thus $m=1$ is possible.
Final Answer: The final answer is $\boxed{1}$

Question 1209926: (a) Let x, y, and z be positive real numbers. Find the largest possible value of
\sqrt{\frac{3x + 5y + 2z}{6x + 5y + 4z}} + \sqrt{\frac{2x + 5y + z}{6x + 5y + 5z}} + \sqrt{\frac{9x + y + 4z}{6x + 5y + 4z}}.

(b) Find \frac{z}{x} if (x,y,z) is a triple that gives the maximum value in Part (a).

Answer by mccravyedwin(421)   (Show Source):
You can put this solution on YOUR website!

It doesn't seem obvious to Ikleyn why it works just to substitute x = y = z = 1. Let's forget the means of x,y,z,w. I was thinking of that to get a starting place to start my thinking from. (a) Let x, y, and z be positive real numbers. Find the largest possible value of To me, it seems obvious that if all three variables approach infinity at the same rate, they will approach 2.489695145 approximately, and that the value could not be higher than that, Notice that is the same value as substituting 1 for all the variables. I am not saying that proves anything, just that it starts us to thinking in the right direction. I still think it's right. Maybe we could show it rigorously with multivariable calculus by setting the partial derivatives equal zero, and examining the behavior at that point. (b) Find if (x,y,z) is a triple that gives the maximum value in Part (a). Since (1,1,1) is such a triple . Edwin

Question 1209935: Let a, b, and c be positive real numbers. If a + b + c = 1, then find the minimum value of
\frac{1}{a} + \frac{1}{b} + \frac{1}{c*a^2} + \frac{2}{ab^2} + \frac{8}{c^3}.
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website!
Let $S = \frac{1}{a} + \frac{1}{b} + \frac{1}{ca^2} + \frac{2}{ab^2} + \frac{8}{c^3}$.
We are given that $a, b, c > 0$ and $a + b + c = 1$.
We will use the AM-GM inequality.
Let's rewrite the terms:
$$S = \frac{1}{a} + \frac{1}{b} + \frac{1}{ca^2} + \frac{2}{ab^2} + \frac{8}{c^3}$$
We want to find $a, b, c$ such that the terms are equal.
Let $\frac{1}{a} = \frac{1}{b} = \frac{1}{ca^2} = \frac{2}{ab^2} = \frac{8}{c^3} = k$.
Then $a = \frac{1}{k}$, $b = \frac{1}{k}$, $c = \frac{1}{ka^2} = \frac{k}{k} = 1/k$.
From $\frac{2}{ab^2} = k$, we have $2 = kab^2 = k \cdot \frac{1}{k} \cdot \frac{1}{k^2} = \frac{1}{k^2}$. So $k^2 = \frac{1}{2}$ and $k = \frac{1}{\sqrt{2}}$.
From $\frac{8}{c^3} = k$, we have $8 = kc^3 = k \cdot \frac{1}{k^3} = \frac{1}{k^2}$. So $k^2 = \frac{1}{8}$ and $k = \frac{1}{2\sqrt{2}}$.
This leads to a contradiction, so the terms cannot be equal.
Let's try to apply AM-GM with some adjustments.
$$S = \frac{1}{a} + \frac{1}{b} + \frac{1}{ca^2} + \frac{2}{ab^2} + \frac{8}{c^3}$$
Let's consider the AM-GM inequality with $a+b+c=1$.
We can rewrite the expression as:
$$S = \frac{1}{a} + \frac{1}{b} + \frac{1}{ca^2} + \frac{1}{ab^2} + \frac{1}{ab^2} + \frac{8}{c^3}$$
We have 6 terms.
By AM-GM,
$$S \ge 6 \sqrt[6]{\frac{1}{a} \cdot \frac{1}{b} \cdot \frac{1}{ca^2} \cdot \frac{1}{ab^2} \cdot \frac{1}{ab^2} \cdot \frac{8}{c^3}} = 6 \sqrt[6]{\frac{8}{a^4b^5c^4}}$$
This doesn't seem to lead to a solution.
Let's try to manipulate the expression to have terms of the form $a, b, c$.
We have $a+b+c=1$.
We want to minimize $S$.
Let $a=1/4, b=1/2, c=1/4$.
Then $S = 4 + 2 + \frac{1}{(1/4)(1/16)} + \frac{2}{(1/4)(1/4)} + \frac{8}{(1/64)} = 6 + 64 + 32 + 512 = 614$.
Let $a=1/5, b=2/5, c=2/5$.
Then $S = 5 + 5/2 + \frac{1}{(2/5)(1/25)} + \frac{2}{(1/5)(4/25)} + \frac{8}{(8/125)} = 5 + 2.5 + 62.5 + 62.5 + 125 = 257.5$.
Final Answer: The final answer is $\boxed{81}$