Understanding if you come up with an equation or an identity

[Giuseppe_97]

Hi guys!
When I'm doing math problems with multiples variables and I have to build up equations, I often come up with identities rather than the variable equal to a value. Is there anyway to understand how we have to build up the equations without obtaining an identity at the end? Thank you.

 

[SteamKing]

It's not clear what you are talking about. An example, perhaps?

 

[Giuseppe_97]

For example the horse-buying problem from Fibonacci.
"Two men have money to buy some horses. The first horse costs x money and the second horse costs x+2 money. The first man says:'If you lend me 1/3 of your money I can buy the first horse.' The second man says:'If you lend me 1/4 of your money I can buy the second horse'.
Find how much money has each of the men and fine the cost of the horses."
As you see there are 3 variables in this problem but, no matter how I algebraically manipulate them I always obtain an identity. Given this, I would like to know if there is a general rule regarding this.

 

[willem2]

As you see there are 3 variables in this problem but, no matter how I algebraically manipulate them I always obtain an identity. Given this, I would like to know if there is a general rule regarding this.

The problem is that there are 3 variables and only 2 equations, so there are multiple solutions, even if you restrict the amounts of money to integers and positive numbers.
The best you can do is to express two of the variables as a function of the third.

 

[symbolipoint]

Hi guys!
When I'm doing math problems with multiples variables and I have to build up equations, I often come up with identities rather than the variable equal to a value. Is there anyway to understand how we have to build up the equations without obtaining an identity at the end? Thank you.

You would usually not focus on finding categories of equations as either identities or just equations. You might use related identities in your problem if you recognize their importance for the problem. You would look to analyze and assign variables to any or all quantities, make drawings if useful, build expressions which fit your problem and numbers, build any equations which relate the numeric expressions. About identities, you choose the ones which will help you understand your problem. Finally, you solve your single equation for one variable or you solve the system of equations for all unknown variables. You need to recognize what and when particular identities are useful or needed. You use identities when they help you analyze or solve a problem.

 

[Stephen Tashi]

I often come up with identities rather than the variable equal to a value.

An equation such as x + 2 = A + B/3 is not an "identity". An identity is an equation that is a correct statement for all possible values of the variables in it. I think that you mean that you come up with equations that have many possible solutions.

Is there anyway to understand how we have to build up the equations without obtaining an identity at the end? Thank you.

As willem2 says, there are some problems that cannot be solved by manipulating equations until you have a single variable equal to a definite value.

The particular example you gave is a "Diophantine" problem and you should look Diophantine Equations, specifically Linear Diophantine equations to see how to solve it.

Are you getting such problems from a course in Number Theory? If so, they won't be solved by the methods you learned in elementary algebra.

 

[Giuseppe_97]

Thanks for the tips guys. Anyway, I didn't express myself very well.
Let's say I have three variables: x,y and z.
Let's say I express every variable in function of x.
By solving the equation though, the x values cancel out and I'm left with 8=8 or 5=5, for example. That's what I meant for identity.
I'm from Italy and next year I'll be in third year of high school. I got this problem from my second year textbook, in the section about systems of equations.

 

[Stephen Tashi]

By solving the equation though, the x values cancel out and I'm left with 8=8 or 5=5, for example. That's what I meant for identity.

You are correct that 8=8 is an identity.

I got this problem from my second year textbook, in the section about systems of equations.

One of topics taught in systems of equations is that some systems have no solutions and some systems have infinitely many solutions. These facts are emphasized when systems of equations are solved in matrix form using elimination or determinants.

The words of the problem resemble those in a set of famous problems by Fibonacci and I have only seen these problems discussed in the context of Diophantine problems, where the solutions are required to be integers.

So it is unclear which topic the problem wished to teach you. Perhaps you should give some other examples.

 

[willem2]

Thanks for the tips guys. Anyway, I didn't express myself very well.
Let's say I have three variables: x,y and z.
Let's say I express every variable in function of x.
By solving the equation though, the x values cancel out and I'm left with 8=8 or 5=5, for example. That's what I meant for identity.
I'm from Italy and next year I'll be in third year of high school. I got this problem from my second year textbook, in the section about systems of equations.

Note that, regardless of the fact that the problem may have one, zero or many solutions, it will always be possible to come up with an identity like 5=5, so if you can derive something like that, it doesn't mean anything.

If there are 2 equations and 3 variables, the best you can do is:

$$(x_1, f(x_1), g(x_1))$$

as the set of solutions