Back To Course

High School Algebra II: Homework Help Resource26 chapters | 281 lessons | 2 flashcard sets

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Systems of linear equations with no solution or infinitely many solutions are special systems of linear equations. This lesson will explore what happens when we try to solve these special systems of linear equations both graphically and algebraically.

Suppose you are sitting in algebra class and a funny thing happens - your teacher catches your attention! This happens when she says that she will cut class short by 5 minutes if anyone in the class can find two numbers, *x* and *y*, such that both of the following equations are true:

*y* = 3*x* + 2

*y* - 3*x* = 0

You get excited, because you recall that this is a system of linear equations, and you know how to solve these!

A **system of linear equations** is a set of linear equations with the same variables. The **solution** to a system of linear equations consists of the values of the variables that make all of the equations in the system true.

To find the solution of a system of linear equations algebraically, we can use **substitution** or **elimination**. For simplicity, we will just use substitution in this lesson, which involves the following steps:

- Solve both equations for the same variable, say
*y*. - Set the two expressions for that variable equal to one another.
- Solve the resulting equation for the other variable, call it
*x*. - Plug the value you find for
*x*into either of the original equations and solve for*y*.

Let's solve this system of equations, so you can go home early! First, we solve both equations for *y*. This only needs to be done for the second equation.

*y* - 3*x* = 0

If we add 3*x* to both sides to solve for *y*, we get:

*y* = 3*x*

Okay, we have *y* = 3*x* + 2 and *y* = 3*x*, so we set 3*x* + 2 equal to 3*x* and solve for *x*.

3x + 2 = 3x |
Subtract 3x from both sides. |

3x + 2 - 3x = 3x - 3x |
Simplify |

2 = 0 | What the heck? |

Huh? That doesn't make any sense, because 2 can't be equal to 0! As it turns out, your teacher pulled a fast one on you. You see, this system is a special type of system that has no solution. Ugh! So much for going home early!

There are two types of special systems of linear equations; those with **no solution**, and those with **infinitely many solutions**. Let's consider what happens when we try to solve these types of systems both graphically and algebraically.

Graphically speaking, the solution to a linear system of equations is the point at which the graphs of the equations in the system intersect. Let's think about this. If a system has no solution, what do you think that means for the graph of the system? You guessed it! Since the system has no solution, the graphs of the equations in the system don't intersect.

See how the graphs of the equations in the system your teacher gave you are parallel, so they don't intersect? This tells us there is no solution to the system.

Now, consider a system with infinitely many solutions. When this is the case, the equations in the system are the same equation, just in different forms, so all of the solutions to one equation are a solution to the other as well. What do we think this means graphically? If you are thinking the graphs of the equations in the system would be the same graph, you are correct! Since they are the same equation, they have the same graph.

We see that systems with infinitely many solutions have infinitely many intersection points.

Let's consider solving algebraically now. Notice that while solving the system with no solution that your teacher gave you, you ran into a statement that made no sense (2 = 0). This is what will always happen when attempting to solve a system with no solution algebraically. You will end up with a statement that is false, such as 5 = 1 or 0 = 4. When this happens, you can stop the solving process and state that the system has no solution.

Now, let's see what happens when we are trying to solve a system with infinitely many solutions using substitution. Consider the example that we graphed.

*y* = 2*x* - 4

*y* = 2(*x* - 2)

The first step is done for us since both equations are solved for *y*, so we set these two expressions equal, and attempt to solve for *x*.

2x - 4 = 2(x - 2) |
Simplify the right hand side |

2x - 4 = 2x - 4 |
Subtract 2x from both sides |

-4 = -4 | Well, duh! |

We end up with the statement -4 = -4, which is always true! In general, when attempting to solve a system with infinitely many solutions, you will run into a statement that is always true, such as 0 = 0 or *x* = *x*.

A **system of linear equations** is a set of linear equations with the same variables. There are two types of special systems of linear equations; those with **no solution** and those with **infinitely many solutions**.

When solving a system graphically, a system with no solution will have equations with graphs that never intersect, and a system with infinitely many solutions will have equations with the same graph, because the equations in the system are actually the same equation.

When solving a system algebraically, a system with no solution will end up with a statement that is always false, such as 0 = 1, and a system with infinitely many solutions will end up with a statement that is always true, such as 3 = 3.

Keeping these rules in mind will come in very handy when solving systems of linear equations. Just remember:

- No solution = No intersection points graphically = A false statement algebraically
- Infinitely many solutions = Infinitely many intersection points graphically = A true statement algebraically

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

BackDid 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
13 in chapter 8 of the course:

Back To Course

High School Algebra II: Homework Help Resource26 chapters | 281 lessons | 2 flashcard sets

- What is a System of Equations? 8:39
- How Do I Use a System of Equations? 9:47
- How to Solve a System of Linear Equations in Two Variables 4:43
- How to Solve a Linear System in Three Variables With a Solution 5:01
- How to Solve a Linear System in Three Variables With No or Infinite Solutions 6:04
- Identity Matrix: Definition & Properties
- System of Equations Word Problem Examples 6:23
- Consistent System of Equations: Definition & Examples 5:38
- Inconsistent System of Equations: Definition & Example 7:37
- Applications of Systems of Equations
- Applications of 2x2 Systems of Equations
- Solving Systems of Three Equations with Elimination
- Solving Special Systems of Linear Equations
- Go to Algebra II - Systems of Linear Equations: Homework Help

- Computer Science 336: Network Forensics
- Computer Science 220: Fundamentals of Routing and Switching
- Global Competency Fundamentals & Applications
- Introduction to the Principles of Project Management
- Praxis Elementary Education: Reading & Language Arts - Applied CKT (7902): Study Guide & Practice
- Practical Applications for Business Ethics
- Practical Applications for Marketing
- Practical Applications for HR Management
- Practical Applications for Organizational Behavior
- Analyzing Texts Using Writing Structures
- MBLEx Prep Product Comparison
- AEPA Prep Product Comparison
- ASCP Prep Product Comparison
- NCE Prep Product Comparison
- TASC Test Score Information
- What is the TASC Test?
- Praxis Prep Product Comparison

- Diclofenac vs. Ibuprofen
- Developing & Managing a High-Quality Library Collection
- Library Space Planning
- Literacy Strategies for Teachers
- Arithmetic Operations in R Programming
- Practical Application: Understanding Employee Behavior
- Positive Global Outcomes of Global Competence
- Practical Application: Color Wheel Infographic
- Quiz & Worksheet - Developing a Learner-Centered Classroom
- Quiz & Worksheet - Technology for Teaching Reading
- Quiz & Worksheet - Pectoralis Major Anatomy
- Quiz & Worksheet - Oral & Written Communication Skills
- Quiz & Worksheet - How to Teach Reading to ELL Students
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies

- Western Civilization I: Help and Review
- AP Psychology: Homeschool Curriculum
- College Macroeconomics: Tutoring Solution
- Psychology: High School
- Introduction to Financial Accounting: Certificate Program
- Continuity in Precalculus: Tutoring Solution
- Algebra II - Inequalities Review: Homework Help
- Quiz & Worksheet - Structure & Function of Adrenal Glands
- Quiz & Worksheet - Divide & Find the Reciprocal of Rational Expressions
- Quiz & Worksheet - The Steps of Family Counseling
- Quiz & Worksheet - Brief Counseling and Therapy
- Quiz & Worksheet - Subduction

- Analyzing the American Short Story: Techniques and Examples
- Legal Moralism: Definition & Examples
- 504 Plans in Michigan
- 2nd Grade Florida Science Standards
- Third Grade Georgia Science Standards
- Word Games for Kids
- Nebraska State Math Standards
- Homeschooling in South Carolina
- What Is Science? - Lesson Plan
- What is on the LSAT?
- Reading Comprehension Lesson Plan
- 4th Grade Georgia Science Standards

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

Browse by subject