Back To Course

ACT Prep: Help and Review44 chapters | 435 lessons | 26 flashcard sets

Instructor:
*Kimberly Hopkins*

Linear combination is a method that is used to solve a system of linear equations. This lesson outlines the three basic ways that linear combination can be used to solve problems.

Have you ever wished that you could make a variable disappear? If so, then linear combination may be for you. **Linear combination** is the process of adding two algebraic equations so that one of the variables is eliminated.

Addition or subtraction can be used to perform a linear combination. Addition is used when the two equations have terms that are exact opposites, and subtraction is used when the two equations have terms that are the same.

There are three basic options for linear combination.

The first option is the easiest. For this option, one of the variables already has coefficients that will cancel when added or subtracted. The first option allows for immediate linear combination.

The second option requires multiplying one of the equations by a constant in order to create a term that can be eliminated using linear combination.

The third option requires multiplying both of the equations by constants in order to combine the equations and eliminate one of the variables.

Now we are going to look at examples of each of these options to get a clearer idea of how they work.

The first option is already in a format that can be cancelled. An example is the system:

2*x* + 2*y* = 10

3*x* - 2*y* = 20

In order to solve this system, we must first look to see if any of the coefficients and variables are exactly the same or exact opposites. 2*y* and -2*y* are exact opposites.

We add the equations by combining our like terms.

2*x* + 3*x* = 5*x*

2*y* + (-2)*y* = 0*y*

10 + 20 = 30.

Our new combined equation is 5*x* = 30.

Next, we solve for *x* by dividing both sides of the equation by 5. Our calculations reveal that *x* = 6.

Lastly, we can substitute 6 for *x* in either of the original equations. Let's substitute 6 in to the first equation.

2(6) + 2*y*= 10 becomes 12 + 2*y* = 10

Subtract 12 from both sides of the equation.

2*y* = -2

Divide both sides of the equation by 2.

*y* = -1

We have now solved the system, and the solution is (6, -1).

The second option requires changing one of the equations. Let's solve the system:

2*x* + 3*y* = 16

3*x* + 6*y* = 30

First, look to see if any of the terms are exactly the same or exact opposites. None of the terms fit that criteria.

Next, look to see if any of the terms are multiples of each other. Notice that 6*y* is a multiple of 3*y*.

We multiply each term in the first equation by 2.

4*x* + 6*y* = 32

Now we have terms that are exactly the same.

We use subtraction to combine the new equation with the second equation.

4*x* - 3*x* = *x*

6*y* - (6)*y* = 0*y*

32 - 30 = 2

Our new combined equation is *x* = 2.

Lastly, we can substitute 2 for *x* in either of the original equations. Let's substitute 2 in to the first equation.

2(2) + 3*y*= 16

4 + 3*y* = 16

Subtract 4 from both sides of the equation.

3*y* = 12

Divide both sides of the equation by 3.

*y* = 4

We have now solved the system, and the solution is (2, 4).

The third option requires changing both equations. Let's look at the system:

3*x* + 5*y* = 22

7*x* - 2*y* = 24

First, look to see if any of the coefficients and variables are exactly the same or exact opposites. None of the coefficients and variables fit that criteria.

Second, look to see if any of the terms are multiples of each other. None of the terms meet that criteria.

Third, look to see if any of the coefficients share a multiple. Notice that 3 and 7 share 21 and 5 and 2 share 10. We can eliminate the *x* or *y* variable. Let's eliminate the *y*.

In order to do this, we must multiply the first equation by 2 and the second equation by 5.

3*x* + 5*y* = 22

6*x* + 10*y* = 44

and

7*x* - 2*y* = 24

35*x* - 10*y* = 120

Now we have terms that are exact opposites. Exact opposites can be eliminated using addition.

We add the equations by combining our like terms.

6*x* + 35*x* = 41*x*

10*y* + (- 10)*y* = 0*y*

44 + 120 = 164

Our new combined equation is 41*x* = 164. Divide both sides of the equation by 41.

*x* = 4

Lastly, we can substitute 4 for *x* in either of the original equations. Let's substitute 4 in to the first equation.

3(4) + 5*y*= 22

12 + 5*y* = 22

Subtract 12 from both sides of the equation.

5*y* = 10

Divide both sides of the equation by 5.

*y* = 2

We have now solved the system, and the solution is (4, 2).

Linear combination is a process that can be used to solve a system of linear equations. Addition and subtraction can be used in the process. There are three basic options for the process:

- One of the variables already has coefficients that will cancel when added or subtracted
- Requires multiplying one of the equations by a constant in order to create a term that can be eliminated using linear combination
- Requires multiplying both of the equations by constants in order to combine the equations and eliminate one of the variables

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackWhat teachers are saying about Study.com

Already registered? Login here for access

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
10 in chapter 13 of the course:

Back To Course

ACT Prep: Help and Review44 chapters | 435 lessons | 26 flashcard sets

- What is a Linear Equation? 7:28
- How to Write a Linear Equation 8:58
- Linear Equations: Intercepts, Standard Form and Graphing 6:38
- What is a System of Equations? 8:39
- How Do I Use a System of Equations? 9:47
- Dependent Events in Math: Definition & Examples
- Direct Variation: Definition, Formula & Examples 7:30
- Equidistant: Definition & Formula
- Introduction to Linear Algebra: Applications & Overview
- Linear Combination: Definition & Examples
- Go to ACT Math - Linear Equations: Help and Review

- Computer Science 335: Mobile Forensics
- Electricity, Physics & Engineering Lesson Plans
- Teaching Economics Lesson Plans
- U.S. Politics & Civics Lesson Plans
- US History - Civil War: Lesson Plans & Resources
- iOS Data Analysis & Recovery
- Acquiring Data from iOS Devices
- Foundations of Digital Forensics
- Introduction to Mobile Forensics
- Examination of iOS Devices
- CNE Prep Product Comparison
- IAAP CAP Prep Product Comparison
- TACHS Prep Product Comparison
- Top 50 Blended Learning High Schools
- EPPP Prep Product Comparison
- NMTA Prep Product Comparison
- Study.com NMTA Scholarship: Application Form & Information

- History of Sparta
- Realistic vs Optimistic Thinking
- How Language Reflects Culture & Affects Meaning
- Logical Thinking & Reasoning Questions: Lesson for Kids
- Mindful Listening Activities
- Marine Science Project Ideas for High School
- Renaissance Project Ideas for High School
- Quiz & Worksheet - Frontalis Muscle
- Octopus Diet: Quiz & Worksheet for Kids
- Quiz & Worksheet - Fezziwig in A Christmas Carol
- Quiz & Worksheet - Dolphin Mating & Reproduction
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies
- 9th Grade Math Worksheets & Printables
- What is Project-Based Learning? | PBL Ideas & Lesson Plans

- NY Regents Exam - US History and Government: Help and Review
- Physical Science for Teachers: Professional Development
- GRE Biology: Study Guide & Test Prep
- HiSET Language Arts - Reading: Prep and Practice
- Middle School Life Science: Help and Review
- Additional CLEP Calculus Flashcards
- Families & Family Issues
- Quiz & Worksheet - Invisible Hand in Economics
- Quiz & Worksheet - Communication Channels in an Organization
- Quiz & Worksheet - Carotenoids
- Quiz & Worksheet - Reducing Agent
- Quiz & Worksheet - The Soldier by Rupert Brooke

- Incomplete Dominance: Definition & Example
- The Intelligent Investor by Benjamin Graham Summary
- Minnesota Science Standards
- Is the SAT a Standardized Test?
- What Are Good SAT Scores?
- Average PSAT Score
- Earth Day Project Ideas
- 2nd Grade Science Projects
- Oregon Science Standards for 3rd Grade
- Transition Words Lesson Plan
- Cool Science Facts
- Homeschooling in Idaho

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject