Systems of Linear Equations

Learning Objectives

In this section, you will learn how to: 

• Identify whether substitution or combination is more efficient for solving a given system of linear equations 
• Apply the techniques of substitution and combination 
• Apply the Kaplan Methods for Problem Solving and Data Sufficiency to questions involving systems of linear equations

In general, if a problem has multiple variables and you want to find unique numerical values for all of the variables, you will need as many distinct equations as you have variables. If you are given two distinct equations with two variables, you can combine the equations to obtain a unique solution set. Don’t be intimidated by calculating values, since usually—though not always—the GMAT will give integer answers for the variables. Focus instead on looking for opportunities to combine equations. The GMAT rewards those who find clever combinations with quick solutions.

Note that the word distinct means that each equation must provide new, different information. In other words, each additional equation must contain information you couldn’t have derived using the equation(s) you already have.

Two Approaches to Solving Systems of Linear Equations

There are two commonly used ways to solve a system of linear equations: substitution and combination. Some systems of equations will be more efficiently solved with substitution and others with combination. Learn both approaches so you are ready to use either one on the questions you see on Test Day.

Substitution

Isolate one variable in one equation. Then plug the expression that it equals into its place in the other equation.

Example: Find the values of m and n if m = 4n + 2 and 3m + 2n = 20.

1.	You know that m = 4n + 2. Substitute 4n + 2 for m in the second equation.	3(4n + 2) + 2n = 20
2.	Solve for n.
3.	To find the value of m, substitute 1 for n in the first equation and solve.	m = 4n + 2 m = 4(1) + 2 m = 4 + 2 = 6

Combination

Add or subtract whole equations from each other to eliminate a variable.

Example: Find the values of x and y if 4x + 3y = 27 and 3x − 6y = −21.

There’s no obvious isolation/substitution to be done, since all variables have coefficients. But if you multiplied the first equation by 2, you’d be able to get rid of the y’s.

Now add the new equation to the second equation, carefully lining them up to combine like terms:

Divide by 11: x = 3

This value can now be substituted back into either equation to yield the other value.

In-Format Question: Systems of Linear Equations on the GMAT

Now let’s use the Kaplan Method on a Problem Solving question dealing with systems of linear equations:

1. If x + y = 5y − 13 and x − y = 5, then x =
   1. 11
   2. 12
   3. 13
   4. 14
   5. 15

Step 1: Analyze the Question

You have two distinct linear equations and two variables, so you will be able to solve for x. Remember that there are two techniques for solving distinct linear equations: (1) substitution and (2) combination. Since these equations look easy to simplify, substituting one equation into the other will be the most efficient approach here.

Step 2: State the Task

Substitute one linear equation into the other to eliminate y and solve for x.

Step 3: Approach Strategically

Simply the first equation:

Simplify the second equation:

Substitute the second equation into the first and solve for x:

Choice (A) is correct.

Step 4: Confirm Your Answer

You can confirm your work by substituting x = 11 into either equation to find a value for y. This value for y can then be substituted into the other equation to ensure that x = 11.

Takeaways: Systems Of Linear Equations

• In a system of equations with n distinct variables, you must have at least n distinct linear equations to be able to solve for each variable.
• Use one of the following two ways to solve a system of linear equations:
  • Substitution: Solve one equation for one of the variables and substitute that value into the other equation.
  • ·   Combination: Add or subtract one equation from the other to cancel out one of the variables.

Practice Set: Systems of Linear Equations on the GMAT 

                 

9. A photographer has visited exactly 42 countries, all of which have been in Africa, Europe, or Asia. If the photographer has been to 6 more countries in Europe than in Asia, and twice as many in Africa as in Europe, how many countries in Asia has the photographer visited?
   1. 4
   2. 6
   3. 12
   4. 18
   5. 24
10. Paula has a collection of used books worth $104. How many are histories?
    1. Paula has only mysteries and histories; each mystery is worth $1, and each history is worth $25.
    2. Paula has exactly 4 mysteries in her collection that together are worth a total of $4.
11. In 8 years, Leonard will be twice as many years old as Mikala is now. If Leonard were twice as old as he was 2 years ago, and if Mikala were five times as old as she was 2 years ago, the sum of their ages would be 51. What will be the sum of their ages in 3 years?
    1. 12
    2. 14
    3. 19
    4. 22
    5. 25