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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

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13 in chapter 8 of the course:

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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

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