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2 Variables 1 Equation

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2 Variables 1 Equation

by NeilWatson » Sun Aug 03, 2014 11:12 am
My question is regarding the concept of 2 variables with 1 equation. I know that in general, once you have an equation with 2 variables, there are is an infinite possibility for the value of those 2 variables but under certain circumstances, the equation can be solved.

Is it the case that if the variables have no integer restrictions, then there is an infinite possibility, but if they are limited to just positive decimals and numbers, then there is only one possibility?

For example, some problems involving dollars and cents give 2 variables that represent the cost of certain items, and it is in fact possible to solve.

Any clarification?
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Source: — Data Sufficiency |

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by [email protected] » Sun Aug 03, 2014 11:23 am
Hi NeilWatson,

If you have a general algebra equation with two variables and NO restrictions....

eg. X + Y = 10

Then are an infinite number of solutions.
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However, the GMAT will occasionally present an equation with two variables and SOME RESTRICTIONS....

eg. X + Y = 10 and both variables are odd numbers greater than 3

With the extra restrictions (odd numbers, greater than 3), then we can deduce that X and Y are both 5.
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The situation that you're describing is more likely to occur in a story problem that is built around limited options. The tricky part is that you might not realize just how limited the options are....

eg. Apples cost 17 cents each, Bananas cost 25 cents each and a shopper spent $2.34 on apples and bananas. How many apples did she buy?

You could set up an equation....

.17A + .25B = 2.34

This might "look" like it has multiple answers, but it only has 1. (A=2, B=8)
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When dealing with any of these situations, it mays to be suspicious. GMAT questions are written with patterns in mind. The writer CHOSE to make the apples cost 17 cents. WHY? The total was $2.34 - that's a WEIRD number. WHY did the writer choose that number? It's to test you on your ability to spot patterns, even when they're not necessarily obvious.

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by Brent@GMATPrepNow » Sun Aug 03, 2014 11:41 am
NeilWatson wrote:My question is regarding the concept of 2 variables with 1 equation. I know that in general, once you have an equation with 2 variables, there are is an infinite possibility for the value of those 2 variables but under certain circumstances, the equation can be solved.

Is it the case that if the variables have no integer restrictions, then there is an infinite possibility, but if they are limited to just positive decimals and numbers, then there is only one possibility?

For example, some problems involving dollars and cents give 2 variables that represent the cost of certain items, and it is in fact possible to solve.

Any clarification?
Great question.

When there are no restrictions on the two variables, then you are correct - the number of solutions is infinite.

However, we can't say that the opposite true. That is, we can't say that imposing restrictions on the variables will yield only one solution. For example, even if we say that x and y are integers, the equation x + y = 7 still has an infinite number of solutions.

Here's the best I can do with regard to a "rule" regarding 1 linear equation with 2 variables: If there are restrictions on the variables, there MIGHT be only one solution.

A proviso: When answering Data Sufficiency questions where there are no restrictions on the variables, some students may accidentally conclude that 1 equation with 2 variables is not sufficient when it actually is sufficient. Consider the target question: "What is the value of x + y?"
Statement 1: 2x + 2y = 6
Here, we have 1 equation with 2 variables, but statement 1 is still sufficient because the target question does not ask us to find the individual values of x and y.

Here's a questions to practice with: https://www.beatthegmat.com/og-13-132-t117594.html


Aside: The whole 1 equation 2 variables thing is a common GMAT trap (along with other common traps). For more information about this and other traps you can watch the following free videos:
https://www.gmatprepnow.com/module/gmat- ... cy?id=1105
https://www.gmatprepnow.com/module/gmat- ... cy?id=1106
https://www.gmatprepnow.com/module/gmat- ... cy?id=1107

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by GMATinsight » Mon Aug 04, 2014 6:12 am
Here is an example of such Question from OG-12 which gives us answer by obtaining unique solution from one equation with two unknowns in constrained scenario

Question : Joanna bought only $0.15 stamps and $0.29 stamps.How many $0.15 stamps did she buy?

(1) She bought $4.40 worth of stamps.
(2) She bought an equal number of $0.15 stamps and $0.29 stamps.
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by GMATinsight » Mon Aug 04, 2014 6:16 am
Question : Joanna bought only $0.15 stamps and $0.29 stamps.How many $0.15 stamps did she buy?

(1) She bought $4.40 worth of stamps.
(2) She bought an equal number of $0.15 stamps and $0.29 stamps.

if x is the number of $0.15 stamps and y is the number of $0.29 stamps.

Question : Determine the value of x

Solution :

Statement 1) Given that 15x + 29y = 440, then 29y = 440 - 15x.

Because x is an integer, 440 - 15x = 5(88 - 3x) is a multiple of 5.
Therefore, 29y must be a multiple of 5, from which it follows that y must be a multiple of 5.

Hence, the value of y must be among the numbers 0, 5, 10, 15, etc.

To more efficiently test these values of y, note that
15x = 440 - 29y, and hence 440 - 29y must be a multiple of 15, or equivalently,
440 - 29y must be a multiple of both 3 and 5.

By computation, the values of 440 - 29y for y equal to 0, 5, 10, and 15 are 440, 295, 150, and 5.

Of these, only 150, which corresponds to y = 10, is divisible by 3.

From 15x = 440 - 29y it follows that x = 10
when y = 10. Th erefore, x = 10 and y = 10;

SUFFICIENT.

Statement 2) Although x = y, it is impossible to determine the value of x because there is no information on the total worth of the stamps Joanna bought.

For example, if the total worth, in dollars, was 0.15 + 0.29, then x = 1,
but if the total worth was 2(0.15) + 2(0.29), then x = 2;
NOT Sufficient.

Answer: Option A
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