How to tell when an equation with two variables has only one solution?

Question

I have notice that it is often the case on DS case questions that the test writers will give you an equation with two variables as one condition. The other condition will give you one of the variables and you end up falling into the "C trap". Does anyone know if there is a way to tell that an equation with two variables has only one unique solution? If so, is there to quickly generate the solution besides simple guess and test? Thanks in advance.

souvonik2k · Answer

Incase of equation with two variables two equations are required. However, incase only 1 equation is given in DS, check if any other condition is given such as number will be integer or positive or it is within some range, so that it can be easily tested with trial and error. In such cases there is no other method rather than trial and error.

JaeTPU · Answer

If you have two linearly independent equations for two variables, you will be able to solve for both variables.

Linearly independent:

What is the value of  x ?
 1) x+y=2 
 2) x+2y=3

The two equations are linearly independent because they are not equivalent. In other words, you are unable to transform either of the equations into another. Since we need 1) and 2) to get two linearly independent equations, we need both statements to be sufficient. Pick C.

Not linearly independent:

What is the value of  x ?
 1) x+y=2 
 2) 2x+2y+3=7

You can subtract 3 from both sides in 2) to get  2x+2y = 4 , and then divide both sides by 2 to get  x+y=2 . This means that 2) is equivalent to 1), so we actually do not have two separate equations. Statements 1) and 2) combined are insufficient. Pick E.

ccooley · Answer

Are you talking about questions that look like this?

If so, this is called a 'combo' problem - 'combo' refers to the fact that you're solving for a combination of two variables, rather than just one variable. (In this case, it's a combo because the question asks you to solve for 2x + 2y, rather than just x by itself or just y by itself.) You can sometimes answer combo questions without actually solving for the variables separately. There isn't a quick and easy way to tell whether this is possible - you just have to take each statement and see if you can use it to figure out the value of the combo 2x + 2y. In the example problem I gave, you'd take statement 1 and multiply both sides by 2, which would give you the value of 2x + 2y.

AvinashHaran · Answer

If there is only one linear equation with 2 unknown variables and without any condition, it will have infinite solutions. 
In case there is a trap in a question, there would be some other information mentioned such as number will be integer or positive or it is within some range. So we need to follow the condition mentioned and solve the question.

Lets say there are 2 linear equations with 2 unknown variables:
ax+by+c=0 &
dx+ey+f=0
If a/d ≠ b/e, there is only one solution

Ex: 2x+3y=6 & 4x+5y=15
These 2 equations have only one solution, as 2/4 ≠ 3/5

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How to tell when an equation with two variables has only one solution?

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