GMAT Linear Equation Problems

Last Updated on November 22, 2023

If you have just started your GMAT quant prep, chances are you have encountered many algebra-related problems, especially linear equation problems.

In fact, GMAT linear equation problems are at the heart of GMAT quant. So, to be successful in that section, you need to know how to solve linear equation problems like the back of your hand.

Solving linear equations for the values of the variable(s) is extremely important. Once you have finished reading this article, you will have a great understanding of how to simplify and solve linear equations.

Note, however, that this article will not focus on graphing linear equations. Although linear equations are often expressed in slope-intercept form (y = mx + b), solving linear equations by graphing is not a focal point of the GMAT. So, we will not address that discussion in this article.

Here are the topics we’ll cover:

• What Is a Linear Equation?
• Linear Equations with One Variable
  • Solving a Linear Equation with One Variable
  • Example 1: Linear Equation with One Variable
  • Example 2: Linear Equation with One Variable
• Linear Equations with Two Variables
  • Solving Linear Equations with Two Variables
  • Substitution Method with Two-Variable Equations
    • Substitution Practice 1
    • Substitution Practice 2
  • Solving Two-Variable Linear Equations Using Substitution
    • Example 1: Linear Equations with Two Variables
  • Solving Systems of Equations by Elimination
  • Using the Elimination Method
    • Example 1: Elimination Method
    • Example 2: Elimination Method
    • Example 3: Elimination Method
  • Sometimes We Have to Change the Sign of a Term Before Adding Equations
    • Example 1: Change the Sign
• Word Problems Involving Linear Equations
  • Example 1: Word Problem
  • Example 2: Word Problem
  • Example 3: Word Problem
• Key Takeaways
• Frequently Asked Questions (FAQ)
  • What Is the Difference Between a Linear Equation and a Quadratic Equation?
  • How Do You Solve a System of Linear Equations?
  • How Do You Use Linear Equations to Solve Word Problems?
• What’s Next?

To begin, let’s discuss the definition of a linear equation.

What Is a Linear Equation?

Although you do not need to know it explicitly for the GMAT, we will discuss the definition of a linear equation. Keep in mind that what we are saying here is true not only for the GMAT, but also for all linear equations. Specifically, there are three main components of a linear equation. A linear equation will have the following:

• one or more variables
• variable(s) raised only to the first power
• variables that are not multiplied together.

Now, let’s look at a few examples of linear equations. When looking at these examples, ensure that you see that all three criteria above are covered.

1. x + y = 10
2. 2n – 5 = 12
3. z/7 + y = 45

Now, let’s look at a few equations that are not linear equations:

1. n^2 = 64

Since n is raised to the second power, we do not have a linear equation.

2. xy = 24

Since we have two x and y variables multiplying each other, we do not have a linear equation.

KEY FACT:

A linear equation must have one or more variables, variables raised to the first power, and variables that are not multiplied together.

Next, let’s discuss linear equations with just one variable.

Linear Equations with One Variable

Although linear equations can have any number of variables, the most basic ones have one variable. A few examples are:

• 5 + n = 12
• 6x – 10 = 2
• 10z – 3 = 12z + 9

First, notice that the above equations are linear with just one variable. Also, recognize that even if the same variable appears twice in the equation, we still have a single-variable equation. For example, the third equation has 10z and 12z, but we have a single-variable equation, since the variable is the same (z).

Solving a Linear Equation with One Variable

Now that we know how to recognize a single-variable equation, let’s discuss how to solve one. The key when solving any linear equation is isolating, or getting alone, the variable that you want to solve for. When the technique is correctly executed, you will always end up with an equation that looks like the following:

variable = some value

The good news is that when solving linear equations, you can follow a step-by-step method that will work for all one-variable equations. Keep in mind that as your skills advance, you may be able to combine some of the following steps. Now, let’s practice this method with an example below.

If 4x + 5 = 2x + 10, then what is the value of x?

Step 1: First, combine the non-variable terms.

To combine the non-variable terms, we subtract 5 from both sides of the equation. Doing so will give us:

4x = 2x + 5

Step 2: Second, combine the variable terms.

To combine the variable terms, we subtract 2x from both sides of the equation. Doing so will give us:

2x = 5

Step 3 (if applicable): Finally, divide both sides of the equation by the coefficient (number in front) of the variable.

Since the coefficient of 2x is 2, we divide both sides of the equation by 2. Doing so gives us:

x = 5/2

KEY FACT:

To solve a linear equation with one variable, first combine the non-variable terms, then combine the variable terms. Then, if necessary, divide both sides of the equation by the coefficient of the variable.

Now, let’s practice with a couple of examples.

Next, let’s practice one more.

Next, let’s discuss equations with two variables.

Linear Equations with Two Variables

Now that we are clear on equations with one variable, let’s look at a few examples of equations with two variables. So, as you may imagine, the only difference between these two types of linear equations is that two-variable equations have two variables! Some examples are shown below:

• x + y = 24
• 2n – 10 = 4m
• 3z – y = 53

While these are a few examples of two-variable equations, we must keep in mind that usually, to determine the value of the variables in these equations, we would need a second equation with at least one of the same variables only, or a second equation with the same two variables. We refer to these multiple equations containing the same variables as systems of linear equations.

(Note: On the GMAT there are instances in which one equation containing two variables can be solved, even if a second equation is not provided. This situation is covered in more detail in our C-Trap blog.)

Now, let’s discuss solving systems of equations by substitution.

Solving Linear Equations with Two Variables

When solving linear equations with two variables (or a system of equations), we can follow many of the steps we followed for the single-variable equations. However, we have one added step of combining the two equations to obtain a final equation that contains just one variable. One method of getting this format is “substitution.” Let’s discuss that now.

Substitution Method with Two-Variable Equations

The way substitution works with linear equations is the same way substituting works in life! For example, remember when you had a substitute teacher in your high school math class? Your substitute teacher replaced your regular teacher.

In math, substitution means essentially the same thing: we are replacing one variable for another variable or variable expression. We should note that before we can use the substitution method, we need a variable isolated in at least one of the equations. Let’s practice substituting.

Substitution Practice 1

Given the following linear equations, solve for the value of y.

x = 2y (equation 1)

x + y = 10 (equation 2)

Solution:

Since we already have x isolated in equation 1, we can substitute 2y for x (in equation 2). After the substitution, equation 2 looks like this:

2y + y = 10 (equation 2)

Now that the equation has only one variable, we can easily solve for y by using the one-variable methods we previously learned.

3y = 10

y = 10/3

Substitution Practice 2

Given the following linear equations, solve for the value of x.

2x – y = 12 (equation 1)

2x + 3y = 20 (equation 2)

Solution:

Looking at the two equations, we should see that it’s easiest to isolate y in equation 1. We do so by adding y to both sides of the equation. We also need to subtract 12 from both sides of the equation. Equation 1 now looks like this:

2x – 12 = y (equation 1)

Since we now have y isolated, we can substitute 2x – 12 for y (in equation 2). Equation 2 now looks like this:

2x + 3(2x – 12) = 20 (equation 2)

We also can distribute 3 in the parentheses. When we do so, we must multiply both 2x and -12 by 3. So, we have:

2x + 6x – 36 = 20 (equation 2)

We now have a one-variable equation. We can solve for x by using the techniques we learned earlier:

2x + 6x – 36 = 20

8x – 36 = 20

8x = 56

x = 7

KEY FACT:

The substitution method allows you to express one variable in terms of the other in the first equation and then substitute that expression into the second equation.

Now, let’s practice solving two-variable linear equations using the substitution method.

Solving Two-Variable Linear Equations Using Substitution

Now that we’ve had ample practice with substituting a variable or expression from one equation to another, we can use this method to solve linear equations with two variables fully. However, we can’t stress enough how critical it is to work with two-variable equations efficiently. So, just ensure that you hone your process.

Next, let’s discuss solving systems of equations by elimination.

Solving Systems of Equations by Elimination

So far, we have discussed how to use substitution when solving systems of linear equations with two variables. As you may recall, the key to substitution is isolating one variable, allowing its equivalent to be substituted for another variable. However, what happens when our equation gets messy when trying to isolate a variable?

For example, let’s look at the equations 2x + 3y = 14 and 5x – 3y = 8, and attempt to isolate x in the first equation.

2x + 3y = 14

2x = 14 – 3y

x = (14 – 3y)/2

We should immediately see that substituting (14 – 3y)/2 for x in the second equation would be tedious and messy. The good news is that, when you find yourself in this position on the GMAT, you can usually use an optional technique: the elimination method. Let’s now discuss that process.

TTP PRO TIP:

When you are unable to cleanly isolate a variable when dealing with two linear equations, you likely will need to use the elimination method.

Using the Elimination Method

Going back to the previous equations of 2x + 3y = 14 and 5x – 3y = 8, we can see that there is no “clean” way to isolate a variable in either equation.

However, we will be able to eliminate one of the variables by adding the equations. The process of adding equations is such that we add like terms.

For example, to add 2x + 3y = 14 and 5x – 3y = 8, we perform the following steps:

• 2x + 5x = 7x (Add the “x” terms from both equations.)
• 3y – 3y = 0 (Add the “y” terms from both equations.)
• 14 + 8 = 22 (Add the constant terms from both equations.)

After adding the equations together, we are left with:

7x + 0 = 22

7x = 22

x = 22/7

Generally, we now see that the elimination method is far superior to the substitution method in many cases.

KEY FACT:

The elimination method requires that we manipulate one or both linear equations such that when we combine the two manipulated equations, one of the variables is eliminated, allowing us to solve for the value of the remaining variable.

Next, let’s practice with a few examples.

Next, let’s discuss how to use the elimination method when we have terms that we can’t eliminate until we make a sign change.

Sometimes We Have to Change the Sign of a Term Before Adding Equations

In the previous sections, we saw how easy it is to add two equations and eliminate one of the variables. However, there will be times when we think we can use the elimination method, but when we add the two equations together, we find that the variables do not cancel.

For example, let’s look at the equations 4n + 7m = 18 and 2n + 7m = 16. If we were to add these two equations together, we would have:

6n + 14m = 34

As we can see, we did not eliminate any variables! However, that outcome does not mean that the elimination method does not work. Rather, it means that if we were to multiply either equation by -1, and then add the equations, we would then be able to eliminate a variable! So, let’s do that now, and determine a value for n.

-1(4n + 7m = 18)

-4n – 7m = -18

Next, we can add our two equations. Let’s go term by term:

-4n + 2n = -2n

-7m + 7m = 0

-18 + 16 = -2

So, we have:

-2n + 0 = -2

-2n = -2

n = 1

KEY FACT:

If two two-variable equations each contain a variable with the same coefficient, we multiply one of the equations by -1. Then, we add the two equations, and one of the variables will be eliminated, allowing us to solve for the value of the remaining variable.

Let’s try an example.

So far, we have discussed the most literal form of GMAT linear equations. But there are many real-world situations that can be modeled by linear equations. So, in the next section, we will discuss general word problems involving linear equations.

Word Problems Involving Linear Equations

As mentioned at the beginning of the article, algebra and linear equations are at the heart of the GMAT. Linear equations can appear in many different types of GMAT questions. In this section, we will discuss the applications of linear equations in word problems.

When we encounter a word problem, we have two jobs. The first is to translate the relationships we are reading about into mathematical equations. The second is to solve those equations. The good news is that all the skills we learned previously in this article will be relevant for solving word problems.

TTP PRO TIP:

Our knowledge of how to solve systems of linear equations is very important when dealing with word problems.

Let’s practice solving a few word problems.

Let’s practice another.

Let’s try one more example.

Key Takeaways

The GMAT quant section heavily focuses on the algebra topic of linear equations. In this article, we have introduced you to the definition of a linear equation. We have also learned how to solve a basic single-variable equation.

A system of linear equations on the GMAT consists of two equations and two variables. We may be asked to solve for the value of one or both of the variables.

The substitution method can be used if one equation can be easily manipulated to allow us to solve for one variable in terms of the other variable.

The elimination method can be used when the substitution method is cumbersome. When we use this method, we multiply one or both of the equations by appropriate constants, allowing one of the variables to be eliminated when the resulting equations are added.

If both equations contain a variable with the same coefficient, we are able to multiply one equation by -1 to allow us to eliminate one of the variables when the resulting equations are added.

Certain word problems can be efficiently solved by using the methods presented in this article.

Frequently Asked Questions (FAQ)

What Is the Difference Between a Linear Equation and a Quadratic Equation?

A linear equation contains a variable whose highest power is 1. A quadratic equation contains a variable whose highest power is 2.

How Do You Solve a System of Linear Equations?

Generally, you can use either the substitution method or the elimination method, each of which is presented in detail in this article.

How Do You Use Linear Equations to Solve Word Problems?

When we encounter word problems, we have two jobs. First, we translate the relationships we are reading about to linear equations, and second, we solve those equations. The good news is that all the skills we learned in this article will be relevant for solving equations involving word problems.

What’s Next?

Linear equations appear on the GMAT frequently. There is no doubt that mastery of the techniques for solving linear equations and systems of linear equations is crucial. But linear equations are just one of the many concepts that you will encounter on the quant section. To understand the scope of the topics that are on the GMAT, take a look at this overview of all the quant topics on the GMAT. To see how these topics are tested on the GMAT, GMAC offers two free practice exams, which you can access after you create your account.

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