Lesson: 2 Equations with 2 Unknowns

Comment on 2 Equations with 2 Unknowns

Hi Brent, could you offer some insight into one of the questions you have linked above?

Is rst = 1 ?

(1) rs = 1
(2) st = 1

It's clear to me why either statement alone is insufficient.

However, when combining them, I substituted the value or r from statement 1 (r = 1/s) into statement 2... essentially, I got both s and t in terms of r, that is, s = 1/r and t = r.

This makes the product rst = (r)(1/r)(r) = r.
This led me to picking E as the answer. Is this valid? All solutions of this question point towards creating tables with counter examples.
gmat-admin's picture

Question link: https://gmatclub.com/forum/is-rst-90398.html

Your solution is perfectly valid.
However, once you conclude that rst = r, it's important that you confirm that r may or may not equal 1.
If it's the case that r MUST equal 1, then the combined statements are sufficient.
If it's the case that r can be ANY value, then the combined statements are not sufficient.

Hi Brent,

Please do you have any advice on how to improve speed? I usually can solve most problems but the harder ones I solve over 2 minutes. :(

Hi Brent. In the elimination method I don't understand why you decided to add 2x+4x and substract 3y-3y but in the next equation you substracted 6x-6x and 5y-5y? How do we know which calculation to make? (addition or subtraction)
gmat-admin's picture

Our goal when solving systems of equations is to eliminate one of the variables.

If a certain variable has coefficients that are EQUAL in the two equations, then we can eliminate that variable by SUBTRACTING one equation from the other.
Example:
5x + 7y = 19
3x + 7y = 17
In this case, we have the same coefficients of y (+7y and +7y)
This means we can eliminate the y terms by subtracting the bottom equation from the top equation to get: 2x = 2

Conversely, if a certain variable has coefficients that are OPPOSITES in the two equations, then we can eliminate that variable by ADDING the two equations.
Example:
5x - 7y = -11
3x + 7y = 27
In this case, we have the coefficients of y are opposites (-7y and +7y)
This means we can eliminate the y terms by adding the two equations to get: 8x = 16

Does that help?

Hi Brent, for (2t + t-x)/(t-x), could this be simplified to (3t-x)/(t-x) = 3? Thank you.
gmat-admin's picture

You're correct to say that (2t+t-x)/(t-x) can be simplified to (3t-x)/(t-x).
However, it is not correct to say that (3t-x)/(t-x) = 3

Here's a useful property: If a/b = c, then a = bc
For example, since 6/2 = 3, we can also say that 6 = (2)(3)

Now let's apply the same property to your example.
If (3t-x)/(t-x) = 3, then we can also say that 3t-x = 3(t-x)
However, when we expand 3(t-x), we get 3t-3x, not 3t-x

On the other hand, (3t-3x)/(t-x) simplifies to equal 3.
Here is why...
3t-3x can be factored to get: (3t-3x)/(t-x) = 3(t-x)/(t-x), and at this point, the two (t-x) terms cancel out to get 3.

For more on simplifying rational expressions, watch the following video lesson: https://www.gmatprepnow.com/module/gmat-algebra-and-equation-solving/vid...

Thank you! I understand where I went wrong.

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