Need an intuitive explanation of what is actually going on with this question

[navneet9431]

Hi,
I have been working to solve this question-"Suppose,we have two strings A and B of lengths 5 cm and 6 cm respectively.And,we have another piece of string C whose length is 4 cm.Now,we have to make the strings A and B equal in length by cutting the string C and adding it to A and B such that string C is completely used.Find what length of strings should be added to A and B ?"

I solved this question in this way,
Let the length of string added to A be "x" and the length of string added to B be "y".
Then,5+x=6+y
=> x-y=1......(i)
And,x+y=4.....(ii)
On adding (i) and (ii) we get,
=>x=5/2=2.5 and y = 1.5

So,i must say that i have gone on to solve this question blindly(without know exactly what really happens when i add those two equation (i) & (ii)).
Now,i want to know what is really going on when i combine(add) those two equations?Why do i get a reasonable value of 'x' when i add them?Why does combining those equations gives me answer of what length should be added? CAN ANYONE HERE GIVE ME AN INTUITIVE EXPLANATION OF WHAT IS REALLY HAPPENING?*I want to know why adding those two equations can precisely divide 4 cm into such two parts which when added to A and B makes them equal(A=B)?*
I have another doubt,
After solving the question i get these answers- (2.5,1.5).
now what i want to know is that why can't there another set of number exist (X,Y) such that X+Y=4 and 5+X=6+Y ? (Why can't another set of number exist which can satisfy the demands of the question ?)

I will be thankful for help!
Note:I am a high school student and English is my second language.

 

[navneet9431]

Strings total 6+5+4 = 15
Look at it as "string material"
To make 2 equal strings: 15/2 = 7.5
It's that simple :cool:

Sorry! But can you please explain my method?
I know that the method you discussed here is more simpler.


Sent from my Elite Note using Tapatalk

 

[JeffM]

Hi,
I have been working to solve this question-"Suppose,we have two strings A and B of lengths 5 cm and 6 cm respectively.And,we have another piece of string C whose length is 4 cm.Now,we have to make the strings A and B equal in length by cutting the string C and adding it to A and B such that string C is completely used.Find what length of strings should be added to A and B ?"

I solved this question in this way,
Let the length of string added to A be "x" and the length of string added to B be "y".
Then,5+x=6+y
=> x-y=1......(i)
And,x+y=4.....(ii)
On adding (i) and (ii) we get,
=>x=5/2=2.5 and y = 1.5

So,i must say that i have gone on to solve this question blindly(without know exactly what really happens when i add those two equation (i) & (ii)).
Now,i want to know what is really going on when i combine(add) those two equations?Why do i get a reasonable value of 'x' when i add them?Why does combining those equations gives me answer of what length should be added? CAN ANYONE HERE GIVE ME AN INTUITIVE EXPLANATION OF WHAT IS REALLY HAPPENING?*I want to know why adding those two equations can precisely divide 4 cm into such two parts which when added to A and B makes them equal(A=B)?*
I have another doubt,
After solving the question i get these answers- (2.5,1.5).
now what i want to know is that why can't there another set of number exist (X,Y) such that X+Y=4 and 5+X=6+Y ? (Why can't another set of number exist which can satisfy the demands of the question ?)

I will be thankful for help!
Note:I am a high school student and English is my second language.

An equation is a statement that two expressions represent the same number.

\(\displaystyle a + b = c * d\)

means nothing more and nothing less than that whatever number a + b may be, c * d is the same number.

Now if two expressions represent the same number and I add the same amount to both expressions, I get new expressions that represent a new number, but those new expressions still represent only one number.

\(\displaystyle a + b = c * d \iff (a + b) + e = (c * d) + e.\)

But the amount added can be expressed in many different ways. Suppose

\(\displaystyle e = \dfrac{f}{g} \text { and } e = m - n \implies\)

\(\displaystyle \dfrac{f}{g} = m - n.\)

\(\displaystyle \text {But } (a + b) + e = (c * d) + e \implies (a + b) + \dfrac{f}{g} = (c * d) + (m - n).\)

Does all this make sense?

Now we can do this more quickly.

\(\displaystyle a + b = c * d \text { and } \dfrac{f}{g} = m - n \implies (a + b) + \dfrac{f}{g} = (c * d) + ( m - n).\)

We just added two equations. Just remember that expressions represent numbers and that an equation says that two expressions represent the same number. So we can add equations and get a new equation.

Is that helpful?

 

[JeffM]

With respect to your second question.

\(\displaystyle expression_1 = expression_2\)

That says that two expressions represent the same number. That statement may be true all the time, some of the time, or none of the time. In other words, some equations are false, mathematical lies.

\(\displaystyle x = x + 1\)

is an equation that is not true for any real value of x. It is an invalid equation, a mathematical lie.

\(\displaystyle x + x = 24 * \dfrac{x}{12}\)

is an equation that is true for every real value of x. It is a valid equation of a type called an identity. Identities are supposed to be written using a variation of the equal sign:

\(\displaystyle x + x \equiv 24 * \dfrac{x}{2}.\)


\(\displaystyle x^2 - 3x + 2 = 0\)

is is a valid equation that is true only if x = 1 or x = 2.

Notice that there are two values of x that make the equation true.

However, there is a special, very simple type of equation called linear. A valid linear equation is either true for every real number or for a single real number. Beginning algebra works with linear equations because they are simple and of tremendous utility in solving practical problems.

 

[Dr.Peterson]

Hi,
I have been working to solve this question-"Suppose,we have two strings A and B of lengths 5 cm and 6 cm respectively.And,we have another piece of string C whose length is 4 cm.Now,we have to make the strings A and B equal in length by cutting the string C and adding it to A and B such that string C is completely used.Find what length of strings should be added to A and B ?"
...
[1] Now,i want to know what is really going on when i combine(add) those two equations?Why do i get a reasonable value of 'x' when i add them?Why does combining those equations gives me answer of what length should be added? CAN ANYONE HERE GIVE ME AN INTUITIVE EXPLANATION OF WHAT IS REALLY HAPPENING?*I want to know why adding those two equations can precisely divide 4 cm into such two parts which when added to A and B makes them equal(A=B)?*
...
[2] what i want to know is that why can't there another set of number exist (X,Y) such that X+Y=4 and 5+X=6+Y ? (Why can't another set of number exist which can satisfy the demands of the question ?)

First, as to the intuitive explanation, what you did in your solution is to translate the problem into an abstract problem, namely to find the solution of a system of linear equations. There is no reason to expect that the solution of an abstract problem will relate in any way directly to the concrete problem from which you derived it. "What is really happening" is that you are eliminating a variable to produce a single equation in one variable that must be true of any solution of the system. I explained this process, in the context of the graphs of the equations rather than an application, here:

Subtracting One Equation from Another
http://mathforum.org/library/drmath/view/53261.html
​

Now, sometimes we can see some analogy between what you are doing in solving, and the actual meaning of the equations. The equations you added are

x - y=1......(i)
x + y=4.....(ii)
​

The first says that the two lengths to be added to the strings must differ by 1; the second says that their sum must be 4. We might graphically represent the two lengths by strips, like this (pardon the ASCII graphics):

+-----------+-------+
| x | y |
+-----------+-------+
<---------4--------->

Adding the first equation amounts to adding x-y to y, making a total of 2x, which adds 1 to the total:

+-----------+-----------+
| x | y+1=x |
+-----------+-----------+
<-----------5----------->

So we now know that 2x is 5, so x is 5/2.

So it can be explained in terms of the problem, but the whole point of algebra, in a sense, is that by transforming a problem into an abstract one, we don't need to think about the actual meaning of anything until we get back to stating the answer. That is the intuition!

As to your second question, we know that there is only one solution because we know that any system of linear equations has only one solution (barring the special cases) because two lines can only intersect in one place (if they are not parallel or coincident lines).