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High School Algebra II: Homework Help Resource26 chapters | 280 lessons | 1 flashcard set

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Lesson Transcript

Instructor:
*Laura Pennington*

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

In this lesson, we'll review systems of equations, commonly used in mathematical problem solving, while concentrating on one particular type: inconsistent systems of equations.

A **system of equations** is a group of equations with the same variables. For example, consider the following system of equations:

*x* - *y* = -1

3*x* + *y* = 9

This is a system of equations in two variables, namely *x* and *y*. As we know, an equation represents a set of points on a graph. Therefore, a system of equations can be represented graphically as well as algebraically.

The **solution set** of a system of equations consists of all the points of intersection of the equations in the system. These points represent values we can plug in for the variables to make all of the equations true. For example, we see that the solution of the system in our graphic is (2,3), because this is where the two equations intersect. There are three possibilities for solutions of a system of equations:

1.) A finite number of solutions: the equations intersect at a finite number of points. Our example above is an example of this - the equations intersect at exactly one point.

2.) Infinitely many solutions: the equations intersect everywhere (they are the same graph).

3.) No solution: the equations do not intersect at all.

Of the three possibilities for the solutions of a system of equations, one possibility is that the system has no solution. When this is the case, we call the system an **inconsistent system of equations**. Look at the following system:

3*x* + 2*y* = 12

3*x* + 2*y* = 6

Notice that the first equation and the second equation of this system have the same left-hand side. However, their right-hand sides differ. So? This means it's not possible to find numbers, *x* and *y*, to plug into 3*x* + 2*y* to equal two different numbers, namely 6 and 12. Therefore, there is no solution to this system, and it's an inconsistent system of equations.

These types of systems are extremely useful when analyzing a problem. If we represent a problem using a system and the system is inconsistent, then we know there is no solution to that problem. This tells us we have to make adjustments or start over altogether. While that may not be ideal in a real-world setting, it can be incredibly time-saving.

There are a number of ways to identify an inconsistent system. One approach is to logically deduce that there is no solution, as we did before. If you can manipulate the equations to get a common left- or right-hand side, and the other side differs between equations, you can logically deduce that there is no solution to the system.

Another way to identify an inconsistent system is by trying to solve the system algebraically. As you're going through the process of solving the system, you will run into a **contradiction**. That is, you will run into something that doesn't make sense, such as 1 = 2 or 0 = 5. Let's look at our example of an inconsistent system of equations. There are multiple ways to solve a system of equations algebraically, but let's just use substitution for simplicity here.

To use **substitution** to solve this system, use the following steps:

1.) Solve one of the equations for one of the variables, say *x*.

2.) Plug the *x*-value you found in step 1 into the other equation to solve for the remaining variable, *y*. You now have a numerical value for *y*.

3.) Plug the numerical *y*-value found in step 2 into either of the original equations. This will result in an equation with just one variable, *x*. Solve for *x*.

At that point, you have numerical values for both variables, and you have your solution.

Let's try applying this to our example system. First, solve one of the equations for one of the variables: *y*. (If you wanted to solve for *x* instead, that's fine. It doesn't matter which variable you solve for in the first step.)

3x + 2y = 12 |
Subtract 3x from both sides... |

2y = 12 - 3x |
Divide both sides by 2... |

y = 6 - 1.5x |
We've solved for y! |

The next step is to plug our value for *y* into the second equation, and then solve for *x*.

3x + 2y = 6 |
Plug in y = 6 - 3/2 x... |

3x + 2(6 - 3/2 x) = 6 |
Distribute... |

3x + 12 - 3x = 6 |
Combine the x terms... |

12 = 6 | This doesn't make any sense! |

We see that when we try to solve the system, we run into the statement 12 = 6, which is impossible, and thus a contradiction. We can stop the solving process now because the system is clearly inconsistent.

The third approach to identifying an inconsistent system is to graph the equations in the system and observe the graph. As we learned earlier, when a system of equations has no solution, the graphs of the equations in the system do not intersect at all. Again, consider our example. The image shows the graph of the two equations in our system:

Notice that the two lines are parallel, so they never intersect. Thus, the system has no solution and is inconsistent.

Let's apply this to a practical example. Suppose you are trying to make a garden with dimensions such that the length plus the width is 20 feet and the perimeter, or the distance around the entire garden, is 50 feet. If we let *l* = length and *w* = width, this situation can be represented by the following system:

*l* + *w* = 20

2*l* + 2*w* = 50

Let's use substitution to solve this. Solve the first equation for *l* to get *l* = 20 - *w*. Plugging this into the second equation, we have 2(20 - *w*) + 2*w* = 50. We then simplify and attempt to solve for *w*.

2(20 - w) + 2w = 50 |
Distribute |

40 - 2w + 2w = 50 |
The 2w terms cancel out |

40 = 50 | This is nonsense! |

We see that we run into a contradiction, so we can't make a garden with the dimensions specified; we will have to find new dimensions. You see how inconsistent systems can really help with error identification and resolution in real-world situations?

A **system of equations** is a group of equations with the same variables. A **solution set** is the set of all the intersection points of the equations in the system. A solution set can have a finite number of solutions, an infinite number of solutions, or no solution. When the system has no solution, we say that the system is **inconsistent**.

We can identify an inconsistent system graphically when the graphs of the equations in the system don't intersect. We can also identify an inconsistent system logically or algebraically. Logically, if we can manipulate the equations so that they all have one side in common, we can look at the other side; if the other side differs, then the system is inconsistent. Algebraically, we will run into a contradiction as we are solving the system, indicating that the system is inconsistent. Knowing what an inconsistent system is and how to identify one can be very time-saving in real-world situations.

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High School Algebra II: Homework Help Resource26 chapters | 280 lessons | 1 flashcard set

- What is a System of Equations? 8:39
- How Do I Use a System of Equations? 9:47
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