Setting two equations equal to one another

[jgoldstein746]

In what circumstances would I be able to set two equations with variables equal to each other? I believe this is called the transitive property of equality.

What circumstances would I not be able to set two equations equal to each other to find an answer to a problem?

What circumstances would I add or subtract two equations to find my answer? For example, solving for both variables;
-2x+y=1
-x-3y=1

Which method(s) could I use?

Thanks!!!

[RonPurewal]

The best answer to all of these questions is "Experiment with the different methods, and find out."
There's no substitute for a good intuition, and the only way to build such as intuition is to accumulate a decent store of experience working with these problems.

So, when you do practice problems, just try things and see whether they work.

What circumstances would I not be able to set two equations equal to each other to find an answer to a problem?

I'm getting the sense that you're looking for a "rule" that you can just memorize here, rather than thinking through the situation at hand. Not good.
Instead of looking for a whole cache of rules, whose success will never be completely assured, try to develop a sense of the goal in each situation, and the tools you have available to help you reach that goal.

(examples in below posts)

[RonPurewal]

For instance, with pairs of equations, you have the following tools available:
* You can multiply or divide the equations by constants.
* You can add the equations to each other, or subtract them from each other.
* If you have two different things equal to the same quantity, then you can set them equal (as mentioned above).

In the situation you gave here:

-2x+y=1
-x-3y=1

* If your goal is to find y -- entailing that you want to get rid of x -- you can multiply the second equation by -2 on both sides, and then add the resulting equations.
* If your goal is to find x -- entailing that you want to get rid of y -- then you can multiply the first equation by 3 on both sides, and then add the resulting equations.
* If you want the expression (-3x - 2y) (or the expression 3x + 2y, which is just the opposite of that one), then you can just add the two equations together exactly as ghiven.
* If you want the expression x - 4y, then you can subtract the first equation from the second one.

You asked about finding both variables, but that will NEVER be the goal of a gmat problem. (GMAT problems can only ask for one quantity per problem.)
Finding both variables would be relevant only in classic "school-type" homework, which is not the same skill set you need to develop for a focus-based test like this one.

[RonPurewal]

In what circumstances would I be able to set two equations with variables equal to each other? I believe this is called the transitive property of equality.

The issue, here, is that you lose some information when you do this. For instance, if x = y and y = z, then you can make x = z -- but then you lose the whole idea that doth of these variables are also equal to y. So, be careful.

So, my best answer is this:

* If you have two expressions that both equal the same thing, you can always set them equal to each other.

* But, you should beware the resultant loss of information.

* As for whether that loss of information is acceptable, you should consider the FOCUS of the problem.
(Remember, a problem will never ask for everything in a particular situation. It will ask only for one quantity, or about one yes/no aspect of the problem.)
If you are losing information that is irrelevant to the focus of the problem, then that's fine.

[jgoldstein746]

Thank you for all of the replies. They have been very helpful!

[RonPurewal]

You're welcome.