This browser is too old to run Achievable.

Update it, or open this page in a current version of Chrome, Safari, Firefox or Edge. On an iPhone or iPad, update iOS: every browser there runs on the version of Safari that comes with it.

Achievable logoAchievable logo
ASVAB
Sign in
Sign up
Purchase
Textbook
Practice exams
Support
How it works
Resources
Exam catalog
Mountain with a flag at the peak
Textbook
Introduction
1. Word Knowledge
2. Mathematics Knowledge
2.1 Foundational math
2.2 Pre-algebra
2.3 Algebra I
2.4 Geometry basics
2.5 Algebra II
2.5.1 FOIL and multiplying polynomials
2.5.2 Systems of equations
2.5.3 Functions, complex numbers, and consecutive integers
2.5.4 Mean, median, mode, range, and factorials
2.6 Advanced geometry
2.7 Math strategies
3. Paragraph Comprehension
4. Arithmetic Reasoning
5. General Science
6. Mechanical Comprehension
7. Electronics Information
8. Shop Information
9. Auto Information
10. Assembling Objects
Wrapping up
Achievable logoAchievable logo
2.5.2 Systems of equations
Achievable ASVAB
2. Mathematics Knowledge
2.5. Algebra II

Systems of equations

5 min read
Font
Discuss
Share
Feedback
Definitions
System of equations
A set of multiple equations with multiple variables

A system of equations is a set of two or more equations with multiple variables you need to identify. The most common type of system problem on the ASVAB is a two-variable system with two equations.

For example, let’s say we have the system:

2x+5y=−1

−3x−10y=4

We want to solve for the values of both x and y.

The answer to this system is x=2 and y=−1. A solution is the pair of values that makes every equation in the system true at the same time: 2(2)+5(−1)=−1 and −3(2)−10(−1)=4. But how did we get there?

There are multiple ways to solve a systems problem, but let’s look at the main two: elimination and substitution.

Elimination method

Definitions
Elimination
Cancelling out a variable in a system of equations

The elimination method is a way to solve a system by getting rid of one variable so you can solve for the other. We do this by lining up the equations and adding or subtracting them until one variable cancels out.

Signs in elimination: Once the coefficients match, look at their signs. If they have the same sign, subtract the second equation from the first. If they have opposite signs, add the two equations.

Example:

To use elimination, we line up the equations vertically and then either add or subtract them to eliminate one of the variables. So let’s first line them up.

2x+5y=−1

−3x−10y=4

Our goal is to get the coefficients for either x or y to match so we can eliminate that variable. For example, we can eliminate y by getting both equations to have a +/−10y term.

In this case, we can get the top equation with 5y to become 10y by multiplying by 2. Be cautious! Whenever we multiply one value in an equation, we must multiply each value in that equation by the same amount.

2(2x+5y=−1)→(4x+10y=−2)

Now line them up:

4x+10y=−2

−3x−10y=4

The y terms now have the same value with opposite signs. Because their signs are opposite, we’ll add both equations to eliminate y.

4x+10y​=−2

−3x−10y​=4

4x−3x=−2+4

x=2

Now we know the value of one variable.

To find y, plug x=2 into one of the original equations:

2x+5y=−1

2(2)+5y=−1

4+5y=−1

5y=−5

y=−1

We’ve found both values: x=2 and y=−1.

Knowledge check:

Solve using elimination:

x+2yx+y​=8=5​

(spoiler)

The x terms have the same sign, so subtract: (x+2y)−(x+y)=8−5, so y=3. Then x+3=5, so x=2.

Substitution method

Definitions
Substitution
Isolating and replacing a variable in a system of equations

As an alternative to elimination, we can use substitution.

To use substitution, we solve one equation for one variable and then plug that expression into the other equation.

Example:

2x+4y=14

5x−2y=−25

First, isolate x in the first equation:

2x+4y=14

2x=−4y+14

x=−2y+7

Now replace x in the second equation:

5x−2y=−25

5(−2y+7)−2y=−25

Now we have one variable:

−10y+35−2y=−25

−12y=−60

y=5

Now plug y=5 back in:

x=−2y+7

x=−2(5)+7

x=−10+7

x=−3

So the solution is x=−3 and y=5.

Knowledge check:

Solve using substitution:

x2x+y​=y+1=8​

(spoiler)

x=3,y=2

When to use each method

Some people prefer one method over the other, while others choose based on the problem.

Here are some quick tips:

  • If one variable is easy to isolate, use substitution
  • If coefficients already match (or are easy to match), use elimination
  • If only one equation needs multiplying, elimination is usually easier
  • If both equations need multiplying, substitution may be simpler

Check the choices: On a multiple-choice question, you can plug each answer pair into both equations. The correct choice is the one that makes both true; a pair that fits only one equation is a trap.

Knowledge check:

Which method would be easiest to use for this problem?

x3x+y​=2y−3=10​

(spoiler)

Substitution, because one variable is already isolated.

Systems of Equations Basics

  • System = two+ equations with multiple variables
  • Solution = variable values that satisfy all equations simultaneously
  • Most common ASVAB type: two-variable, two-equation system

Elimination Method

  • Line up equations vertically; manipulate so one variable’s coefficients match
  • Must multiply every term in an equation by the same factor when scaling
  • Same signs → subtract equations; opposite signs → add equations to cancel variable
  • After solving for one variable, substitute back into an original equation to find the other

Substitution Method

  • Isolate one variable in one equation (solve for x or y)
  • Plug that expression into the other equation to reduce to one variable
  • Solve for remaining variable, then back-substitute to find the first

Choosing a Method & Test Strategy

  • Use substitution if a variable is already (or easily) isolated
  • Use elimination if coefficients match or only one equation needs multiplying
  • If both equations require multiplying, substitution is often simpler
  • On multiple-choice questions, plug answer pairs into both equations — correct answer satisfies both (watch for traps that satisfy only one)

Sign up for free to take 7 quiz questions on this topic

Previous
Next  | 2.5.3 Functions, complex numbers, and consecutive integers
All rights reserved ©2016 - 2026 Achievable, Inc.

Systems of equations

Definitions
System of equations
A set of multiple equations with multiple variables

A system of equations is a set of two or more equations with multiple variables you need to identify. The most common type of system problem on the ASVAB is a two-variable system with two equations.

For example, let’s say we have the system:

2x+5y=−1

−3x−10y=4

We want to solve for the values of both x and y.

The answer to this system is x=2 and y=−1. A solution is the pair of values that makes every equation in the system true at the same time: 2(2)+5(−1)=−1 and −3(2)−10(−1)=4. But how did we get there?

There are multiple ways to solve a systems problem, but let’s look at the main two: elimination and substitution.

Elimination method

Definitions
Elimination
Cancelling out a variable in a system of equations

The elimination method is a way to solve a system by getting rid of one variable so you can solve for the other. We do this by lining up the equations and adding or subtracting them until one variable cancels out.

Signs in elimination: Once the coefficients match, look at their signs. If they have the same sign, subtract the second equation from the first. If they have opposite signs, add the two equations.

Example:

To use elimination, we line up the equations vertically and then either add or subtract them to eliminate one of the variables. So let’s first line them up.

2x+5y=−1

−3x−10y=4

Our goal is to get the coefficients for either x or y to match so we can eliminate that variable. For example, we can eliminate y by getting both equations to have a +/−10y term.

In this case, we can get the top equation with 5y to become 10y by multiplying by 2. Be cautious! Whenever we multiply one value in an equation, we must multiply each value in that equation by the same amount.

2(2x+5y=−1)→(4x+10y=−2)

Now line them up:

4x+10y=−2

−3x−10y=4

The y terms now have the same value with opposite signs. Because their signs are opposite, we’ll add both equations to eliminate y.

4x+10y​=−2

−3x−10y​=4

4x−3x=−2+4

x=2

Now we know the value of one variable.

To find y, plug x=2 into one of the original equations:

2x+5y=−1

2(2)+5y=−1

4+5y=−1

5y=−5

y=−1

We’ve found both values: x=2 and y=−1.

Knowledge check:

Solve using elimination:

x+2yx+y​=8=5​

(spoiler)

The x terms have the same sign, so subtract: (x+2y)−(x+y)=8−5, so y=3. Then x+3=5, so x=2.

Substitution method

Definitions
Substitution
Isolating and replacing a variable in a system of equations

As an alternative to elimination, we can use substitution.

To use substitution, we solve one equation for one variable and then plug that expression into the other equation.

Example:

2x+4y=14

5x−2y=−25

First, isolate x in the first equation:

2x+4y=14

2x=−4y+14

x=−2y+7

Now replace x in the second equation:

5x−2y=−25

5(−2y+7)−2y=−25

Now we have one variable:

−10y+35−2y=−25

−12y=−60

y=5

Now plug y=5 back in:

x=−2y+7

x=−2(5)+7

x=−10+7

x=−3

So the solution is x=−3 and y=5.

Knowledge check:

Solve using substitution:

x2x+y​=y+1=8​

(spoiler)

x=3,y=2

When to use each method

Some people prefer one method over the other, while others choose based on the problem.

Here are some quick tips:

  • If one variable is easy to isolate, use substitution
  • If coefficients already match (or are easy to match), use elimination
  • If only one equation needs multiplying, elimination is usually easier
  • If both equations need multiplying, substitution may be simpler

Check the choices: On a multiple-choice question, you can plug each answer pair into both equations. The correct choice is the one that makes both true; a pair that fits only one equation is a trap.

Knowledge check:

Which method would be easiest to use for this problem?

x3x+y​=2y−3=10​

(spoiler)

Substitution, because one variable is already isolated.

Key points

Systems of Equations Basics

  • System = two+ equations with multiple variables
  • Solution = variable values that satisfy all equations simultaneously
  • Most common ASVAB type: two-variable, two-equation system

Elimination Method

  • Line up equations vertically; manipulate so one variable’s coefficients match
  • Must multiply every term in an equation by the same factor when scaling
  • Same signs → subtract equations; opposite signs → add equations to cancel variable
  • After solving for one variable, substitute back into an original equation to find the other

Substitution Method

  • Isolate one variable in one equation (solve for x or y)
  • Plug that expression into the other equation to reduce to one variable
  • Solve for remaining variable, then back-substitute to find the first

Choosing a Method & Test Strategy

  • Use substitution if a variable is already (or easily) isolated
  • Use elimination if coefficients match or only one equation needs multiplying
  • If both equations require multiplying, substitution is often simpler
  • On multiple-choice questions, plug answer pairs into both equations — correct answer satisfies both (watch for traps that satisfy only one)

More from Algebra II

  • FOIL and multiplying polynomials
  • Functions, complex numbers, and consecutive integers
  • Mean, median, mode, range, and factorials