Find Two Numbers -- Have Two Equations and Two Unknowns

[nycmathguy]

Homework Statement

Find Two Missing Numbers

Relevant Equations

System of Equations

Two numbers add up to 72. One number is twice the other. Find the numbers.

Let x and y be our two numbers.

Two numbers add to 72.

x + y = 72

One number is twice the other.

I can use x or y for this next set up.

x = 2y

Here is the system:

x + y = 72
x = 2y

You say?

 

[phinds]

You say?

I say solve it and get on with your life.

EDIT: if you don't know it already, the way to check this is not to ask on an internet forum but rather to just SOLVE it and then plug the numbers back into the problem and see if they fit

 

[Mark44]

I say solve it and get on with your life.

+1

 

[Delta2]

I don't understand @phinds, why he shouldn't post this here and ask if he is doing any mistakes? That's what we usually do in the homework forums.

 

[phinds]

I don't understand @phinds, why he shouldn't post this here and ask if he is doing any mistakes? That's what we usually do in the homework forums.

Sure, but first he should do what I suggested and then he would know that he didn't NEED to post here. Isn't that what you would have done?

 

[Delta2]

Sure, but first he should do what I suggested and then he would know that he didn't NEED to post here. Isn't that what you would have done?

I guess he wanted a confirmation from us that he formed the right system of equations. And also he might need help on how to solve it.

 

[berkeman]

Yeah, but this post from one of his other threads is a bit worrisome, IMO:

I will post hundreds of word problems here in the coming months.

If most of those are simple ones that he can solve and check for himself, that's not a good use of PF. We're here for when you try your best but get stuck, or when you plug your solutions back into the original word problem they don't work...

 

[Mark44]

Here is the system:
x + y = 72
x = 2y
You say?

I say solve it and get on with your life.

I'm with @phinds here. @nycmathguy has done the heavy lifting of coming up with a system of equations, which is more than 50% of the problem. IMO he should have solved this simple system and checked his solution against the problem statement. If the solution worked in the problem statement, no need to even post the question. If the solution didn't work, then that would have been reason for starting a thread, and we could have helped out.

 

[nycmathguy]

I'm with @phinds here. @nycmathguy has done the heavy lifting of coming up with a system of equations, which is more than 50% of the problem. IMO he should have solved this simple system and checked his solution against the problem statement. If the solution worked in the problem statement, no need to even post the question. If the solution didn't work, then that would have been reason for starting a thread, and we could have helped out.

So, everything I do here is wrong, huh?

 

[Mark44]

So, everything I do here is wrong, huh?

No, that's not what I'm saying. What I am saying is why didn't you take a stab at solving the system of equations you came up with?

 

[phinds]

So, everything I do here is wrong, huh?

It's a serious waste of your time and energy to get snippy about appropriate criticism rather than trying to learn from it. You did a great job of creating the right equations and has been said, that's half (actually I think it's a lot MORE than half) of the battle in solving word problems.

Now you just need to learn to follow through, solve your equations, and check your work. Until you learn to do that you will continue to feel lost w/ word problems.

 

[nycmathguy]

It's a serious waste of your time and energy to get snippy about appropriate criticism rather than trying to learn from it. You did a great job of creating the right equations and has been said, that's half (actually I think it's a lot MORE than half) of the battle in solving word problems.

Now you just need to learn to follow through, solve your equations, and check your work. Until you learn to do that you will continue to feel lost w/ word problems.

Understood. However, did you ever stop to think that perhaps I work a 40-hour overnight shift and sleeping during the day is crucial? Yes, I have the weekend off but weekend hours fly by. My dad is to say that the weekend is an illusion.

 

[willem2]

You post a system of equations,. and then "you say?".
It's really not clear what you mean here?
- Do you mean to ask if this system is correct. (yes)
- Do you really want to say that you have a system, but no idea how to solve it? (you'll probably get some hint about what to do if you ask that)
We won't solve it, if you really need the answer, you can type it in wolfram alpha, or an online graphical calculator.

I understand you made some related posts about word problems, but you really shouldn't count on anyone having read those. People don't know how to react to a post like this.

 

[nycmathguy]

You post a system of equations,. and then "you say?".
It's really not clear what you mean here?
- Do you mean to ask if this system is correct. (yes)
- Do you really want to say that you have a system, but no idea how to solve it? (you'll probably get some hint about what to do if you ask that)
We won't solve it, if you really need the answer, you can type it in wolfram alpha, or an online graphical calculator.

I understand you made some related posts about word problems, but you really shouldn't count on anyone having read those. People don't know how to react to a post like this.

I simply wanted to know if my set up is correct. Naturally, I know how to solve this system of equations.

 

[nycmathguy]

I say solve it and get on with your life.

EDIT: if you don't know it already, the way to check this is not to ask on an internet forum but rather to just SOLVE it and then plug the numbers back into the problem and see if they fit

Here is the system:

x + y = 72
x = 2y

You said to solve it and move on.

Let x + y = 72 be Equation A.

Let x = 2y be Equation B.

Replace x in A by 2y and solve for y.

x + y = 72

2y + y = 72

3y = 72

y = 72/3

y = 24

I now plug y = 24 into either A or B to find x.

I will use Equation B.

x = 2y

x = 2(24)

x = 48

The two missing numbers are 24 and 48.

Evidence:

24 + 48 = 72

72 = 72

Are you happy now? I can't seem to do anything right here, huh? My Homework Statement is wrong. My Relevant Equations is always wrong. What the heck!

 

[nycmathguy]

It's a serious waste of your time and energy to get snippy about appropriate criticism rather than trying to learn from it. You did a great job of creating the right equations and has been said, that's half (actually I think it's a lot MORE than half) of the battle in solving word problems.

Now you just need to learn to follow through, solve your equations, and check your work. Until you learn to do that you will continue to feel lost w/ word problems.

I need to follow through, huh

Ok.

Here is the system again:

x + y = 72
x = 2y

Let x + y = 72 be Equation A.

Let x = 2y be Equation B.

Replace x in A by 2y and solve for y.

x + y = 72

2y + y = 72

3y = 72

y = 72/3

y = 24

I now plug y = 24 into either A or B to find x.

I will use Equation B.

x = 2y

x = 2(24)

x = 48

The two missing numbers are 24 and 48.

Evidence:

24 + 48 = 72

72 = 72

Extension:

I needed to set up a system of equations to find the two missing numbers. Can this be done without setting up a system of equations? Say I came across this problem on a test without knowing anything about a system of equations or how to set it up, what would I need to do to find the missing numbers?

 

[phinds]

I needed to set up a system of equations to find the two missing numbers. Can this be done without setting up a system of equations? Say I came across this problem on a test without knowing anything about a system of equations or how to set it up, what would I need to do to find the missing numbers?

You do need a system of equations, however, once you get used to working on these very simple problems you can often just do them quickly in your head without needing to write anything down.

 

[nycmathguy]

You do need a system of equations, however, once you get used to working on these very simple problems you can often just do them quickly in your head without needing to write anything down.

You said "simple problems" in your reply. You do understand that my threads will increasingly get harder as I go through the textbooks, right? This also applies to word problems as well. Of course, this particular problem can be solved by middle school students. My goal is to learn precalculus through calculus 3. Obviously, the questions are going to get harder. Along the way, I will find time to occasionally post algebra, geometry, trigonometry and calculus word problems. I am slowly going through the books. Rushing leads to more confusion.

 

[phinds]

You said "simple problems" in your reply. You do understand that my threads will increasingly get harder ...

Of course, and as they do the ability to solve them in your head will quickly vanish. I was specific about my comment relating to the simple ones, like the one being discussed.

 

[nycmathguy]

Of course, and as they do the ability to solve them in your head will quickly vanish. I was specific about my comment relating to the simple ones, like the one being discussed.

I get it now.

 

[PeroK]

I simply wanted to know if my set up is correct. Naturally, I know how to solve this system of equations.

Here's a different way to think about things. At an elementary level, it's common to undertake dozens of problems that are all essentially the same. In this case, one number is some multiple of another and they sum to a given total. You repeat the process for dozens of questions and get the idea.

"Real" maths, however, is more about seeing the general pattern and encoding it mathematically. For example:

One number is $a$ times another number and they sum to $b$. What are the numbers?

If we take the two numbers to be $x$ and $y$ we have:

$y = ax$

$x + y = b$

This gives:

$x = \frac {b}{1 + a}$

$y = \frac{ab}{1 + a}$

That, IMHO, is mathematics.

To check things out for your example: $a = 2, b = 72$:

$x = \frac{72}{3} = 24$

$y = \frac{2 \times 72}{3} = 48$

 

[Delta2]

That, IMHO, is mathematics

You just did a generalization of the problem and its solution, why do you think that the generalization is mathematics while the solution for the specified instance of problem is not mathematics?

 

[PeroK]

You just did a generalization of the problem and its solution, why do you think that the generalization is mathematics while the solution for the specified instance of problem is not mathematics?

To master any subject you need the enlightenment - the insights that allow you really to understand what you are doing.

With chess, for example, it's the difference between learning the moves and being able to play the game. With computer programming it's the difference between learning the syntax and being able to put a significant programme together. And, IMO, with mathematics it's the difference between plodding through the same algebraic steps time after time and seeing the underlying algebra.

 

[Delta2]

To master any subject you need the enlightenment - the insights that allow you really to understand what you are doing.

I can't disagree with this but usually to understand the insights (which are generalized insights if i can call them that way, they refer to a generic version of the problem) we have to work through specific instances, examples of the problem , usually one (at least a novice-newcomer) doesn't understand the generalized version unless he works with specific examples first.

 

[nycmathguy]

Here's a different way to think about things. At an elementary level, it's common to undertake dozens of problems that are all essentially the same. In this case, one number is some multiple of another and they sum to a given total. You repeat the process for dozens of questions and get the idea.

"Real" maths, however, is more about seeing the general pattern and encoding it mathematically. For example:

One number is $a$ times another number and they sum to $b$. What are the numbers?

If we take the two numbers to be $x$ and $y$ we have:

$y = ax$

$x + y = b$

This gives:

$x = \frac {b}{1 + a}$

$y = \frac{ab}{1 + a}$

That, IMHO, is mathematics.

To check things out for your example: $a = 2, b = 72$:

$x = \frac{72}{3} = 24$

$y = \frac{2 \times 72}{3} = 48$

Impressive reply. Great math notes for me.

 

[gmax137]

...

This gives:

$x = \frac {b}{1 + a}$

$y = \frac{ab}{1 + a}$

...

The only problem I see with this, is having the student memorize the "formulas" without knowing how to derive them. Then in the quiz, it is "what formula do I use" rather than "how do I solve the system."

 

[nycmathguy]

The only problem I see with this, is having the student memorize the "formulas" without knowing how to derive them. Then in the quiz, it is "what formula do I use" rather than "how do I solve the system."

I hated that back in my student days. Here's a formula: now memorize it. What does the formula mean? Students have zero idea.

 

[gmax137]

My concern is more with a certain type of student, who chooses to ignore (not understand) the derivation, instead choosing to memorize the formulas. There are plenty of homework help threads here that begin "what formula do I use to..."

 

[PeroK]

The only problem I see with this, is having the student memorize the "formulas" without knowing how to derive them. Then in the quiz, it is "what formula do I use" rather than "how do I solve the system."

Your comment astonishes me and perhaps shows the depth of the problem for those who do not think mathematically. You're not supposed to memorise any formulas! You're supposed to recognise that there is a general algebraic method behind all the problems that are essentially the same, but with different numbers. You're supposed to see the algebraic pattern and begin to digest the real purpose of algebra.

The counterargument, which I think this thread shows is valid, is that the longer you soldier on one problem at a time, seeing no connection between one problem and the next, the harder it is ever to escape that cycle. Your brain never makes the underlying connections because you're not looking for them. And by the time someone sits you down and says "now for some real algebra", your brain has been trained not to think like that and it's a shock to the system.

 

[gmax137]

Your comment astonishes me

I think I have been misunderstood, probably because I was not clear on the nature of my concern.

 

[symbolipoint]

I can't disagree with this but usually to understand the insights (which are generalized insights if i can call them that way, they refer to a generic version of the problem) we have to work through specific instances, examples of the problem , usually one (at least a novice-newcomer) doesn't understand the generalized version unless he works with specific examples first.

Regardless of newcomer, almost-newcomer, or whatmore, students will often enough find the SAME exercise but as different examples, using differing values and maybe too diverse situations --but all of the examples being the SAME general form. That is why studying and applying algebra that way is so useful.