Solving GMAT Questions with two linear equations and two unknowns

In order to solve such equations, you need at least 2 distinct equations involving these unknowns.

For example, if we are trying to solve for x and y, we won't be able to solve it using these 2 equations.

2x + y = 14
4x + y - 14 = 14 - y

Why?  Because the two equations on top are the same.  If you simplify the second equation, you get 4x + 2y = 28 which reduces to 2x + y = 14 - the same equation as the first.  If the two equations are the same, then there will be infinitely many values for x and y that will satisfy the equations.  For example, x = 2 and y = 10 satisfies the equation.  So does x = 4 and y = 8.  And so does x = 6 and y = 2.

In order to solve for an actual value of x and y, we need 2 distinct equations.

For example, if we had

2x + y = 14 --------(1)
x - y = 4 ----------(2)

Then from equation (2), we can get x = 4 + y and substitute that into equation (1) to get:
2(4 + y) + y = 14  We can then solve for y.  See if you got y = 2  Once you've got y = 2, you can substitute that into x= 4 + y to get x = 6.

An important lesson here is that you need as many distinct equations as you have variables.  So if you are doing a data sufficiency question, you can sometimes just note that as long as you have two distinct equations, you will be able to solve for x and y.  Disregard or other equivalent equations.

Sometimes, however, its possible for there to be no solution to the set of equations.  This occurs when one side of each equation is the same, but the other side is different.  For example,

2x + y = 14
2x + y = 0

has no solution.  Its like saying that the same steak costs $14 and $0 at the same restaurant.  Of course, the GMAT is not going to make it so obvious that the equations contradict each other. 

Usually, you would have to simplify one of the equations to check if it is

1. the same same as the other equation, in which case, you have infinitely many solutions
2. the same as the other equation on one side, but different on the other side of the equal sign.  In this case, you have no solution
3. a distinct equation, in which case x and y has an exact value.

The GMAT might also require you to translate a word problem into a pair of simultaneous equations to solve.  Let's try this question from Grockit.

A package contains nothing but 35 DVDs and 15 videotapes. What is the total weight, in pounds, of the contents of the package?

(1) Each videotape weighs twice as much as each DVD.
If we let the weight of a DVD be x and a videotape be y, then according to this statement y = 2x
(2) The total weight of 2 of the videotapes and 2 of the DVDs is 1 pound.

Following the previous notation, 2y + 2x = 1

Author : 

Grockit, an online test prep game, is the smartest way to study for your test. It's adaptive, fun and finds the right teacher for you. Grockit’s analytic capabilities and adaptive technology identifies students' strengths and weaknesses, focusing the student's study time.  Students can practice in adaptive solo games, play social learning games with peers, and work with experts that match their specific needs.

Essential GMAT Reading Comprehension Guide

After you read F1GMAT’s Essential GMAT Reading Comprehension Guide, you will:

1. Complete GMAT RC Questions in less than 1 minute and 50 seconds

2. Read Faster

3. Take Notes Effectively

4. Collect and Interpret Facts

5. Speed up Summary Creation

6. Remember Information

7. Question the Author   

8. Learn to answer GMAT Reading Comprehension Title question   

9. Learn to Solve GMAT Reading Comprehension Main Idea Question   

10.Learn to Solve GMAT Reading comprehension inference question   

11. Learn to Solve GMAT Reading Comprehension Detail Questions   

12. Learn to Organize passage in GMAT Reading Comprehension   

13. Learn to Identify style/tone or attitude of the author

14. Learn to Improve GMAT Reading Comprehension Score

Download Essential GMAT Reading Comprehension Guide

Mastering GMAT Critical Reasoning

After you read F1GMAT’s Mastering GMAT Critical Reasoning Guide, you will:

1) Learn to eliminate out of scope answer choices

2) Learn to spot logical fallacies

3) Learn to read questions by focusing on the holy trinity – premise, assumption, and conclusion.

4) Learn to disregard filler information

5) Complete GMAT CR Questions in less than 1 minute and 40 seconds

Download Mastering GMAT Critical Reasoning

Navigate F1GMAT

F1GMAT Services (MBA Applicants) 

MBA Research

MBA Salary(Latest Salary Data)
MBA Admission Interview Tips



MBA Application Essays

GMAT Tutorials

GMAT Question Bank

Top MBA Programs

Get F1GMAT's Newsletters (Best in the Industry)
Included in the Newsletter:

  • Ranking Analysis
  • Post-MBA Salary Trends
  • Post-MBA Job Function & Industry Analysis
  • Post-MBA City Review
  • MBA Application Essay Tips
  • School Specific Essay Tips
  • GMAT Preparation Tips
  • MBA Admission Interview Tips
  • School Specific Interview Tips
  • Funding Guidance and
  • Special Consultation Service (only for Subscribers)

Subscribe to F1GMAT's Newsletter