Combining equations

Question

What is the rule for combining equations? The books I have read say that you can always add or subtract equations in order to cancel out a variable so that you can solve for another one. However, I was reviewing the answer to one question in the OG 13th edition, and it appears that they combined by multiplying. Below is the question and response. Thanks!

Is rst = 1?
(1) rs = l
(2) st = l

[After proving 1) and 2) are not sufficient, consider both statements together.]

We can combine the equations, trying to put rst on one side of the resulting equation.
(rs)(st)=(1)(1)
rs^2t=1
rst=1/s

We cannot isolate rst on one side of the question and have a number by itself on the other side. Therefore, we cannot find a specific value for rst, and we cannot confirm of deny that rst=1.

So, is it always okay to say, okay you can multiple the left side by the left side and the right side by the right side of two equations? I guess in theory that should be doing the same thing to both side of the equation. It just seems weird that that was never suggested in any of the combining equations problems I've seen.

Thanks!

dimochka · Answer

Hi jroach31,

Yes, you can add equations, multiply them, or do anything else to them, as long as you're doing the same thing to each side. Essentially you are working with some statements A,B,C,D, where A=B and C=D. So if A=B, A*C = B*C. But since C=D, you can also write it as A*C = B*D.

Looking at the question you just outlined more generally:
(1) rs = x
(2) st = y

that means that rs * st = x * st (we can always multiply both sides by the same number). But based on (2) st = y, so you can write rs * st = x * st = rs * y = x * y.

Rich.C@EMPOWERgmat.com · Answer

Hi jroach31,

dimochka has re-iterated the immediate point: yes, you can multiply equations, much in the same way that if you do something to "one side" of the equation, you have to do it to the "other side."

Since GMAT questions can be approached in a number of different ways, it's important to be develop multiple skills while you're studying. Certain questions are more susceptible to certain approaches/tactics than others. For example, a typical "system" question (2 variables and 2 unique equations) is usually solved with straight-forward math. The DS question that you listed, however, can be easily solved by TESTing Values.

Is rst = 1?

This is a YES/NO question. We know nothing about the three variables (meaning that they could be ANYTHING).

Fact 1: rs = 1

We know nothing about the value of t….
If t = 1, then rst =1(1) = 1 and the answer is YES
If t = 0, then rst =1(0) = 0 and the answer is NO
Fact 1 is INSUFFICIENT

Fact 2: st = 1

We know nothing about the value of r…
…so we'd have the same situation as in Fact 1 (YES and NO answers)
Fact 2 is INSUFFICIENT

Combined
rs = 1 and st = 1
If r=1, s=1 and t=1, then rst = 1 and the answer to the question is YES
If r=1/2, s = 2 and t = 1/2, then rst = 1/2 and the answer to the question is NO.
Combined INSUFFICIENT

Final Answer: E

Most books tend to showcase the "math" approach in the explanations to questions, but there's usually another way to solve the problem, so keep an open mind to approaches beyond the one offered.

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

GMATGuruNY · Answer

jroach31 wrote:
Is rst = 1?
(1) rs = l
(2) st = l

Each statement alone is clearly insufficient.

Statements combined:
Since rs = 1 and st = 1, rs = st = 1.
Thus:
rs = st
r = t.
It's possible that r=t=1 and s=1, in which case rst = 1.
It's possible that r=t=2 and s=1/2, in which case rst = 2.
INSUFFICIENT.

The correct answer is E.

Alternate approach:
Dividing rs =1 by st = 1, we get:
rs/st = 1/1
r/t = 1
r = t.
From here, we can proceed as I did above.

Alternate approach 2:
Since rs = 1, s = 1/r.
Substituting s = 1/r into st = 1, we get:
(1/r)t = 1
t = r.
From here, we can proceed as I did above.

Alternate approach 3:
Since t is common to both equations, plug in a value for t and solve for r and s.
Plugging in t=1 into rt = 1 and st = 1, we get:
r*1 = 1
r=1.
s*t = 1
s = 1.
In this case, rst = 1*1*1 = 1.

Plugging in t=2 into rt = 1 and st = 1, we get:
r*2 = 1.
r = 1/2.
s*2 = 1
s = 1/2.
In this case, rst = (1/2)(1/2)(2) = 1/2.
INSUFFICIENT.

As Rich has noted, there are many ways to combine equations.

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