Solve each system of equations. If the system has no solution, say that it is inconsistent. {

Solve each system of equations. If the system has no...

Key Concepts

Consistency and Inconsistency

The concepts of consistency and inconsistency refer to whether a system of equations has at least one solution or none. A consistent system has at least one solution, whereas an inconsistent system has no solutions. In the process of solving such a system, if one arrives at a contradiction (like an equation stating 0 equals a nonzero number), it indicates that the system is inconsistent.

System of Linear Equations

A system of linear equations is a collection of one or more linear equations involving the same set of variables. Such systems can have a unique solution, infinitely many solutions, or no solution at all depending on the relationships between the equations. They form the basis of many applications in algebra and are key to understanding more advanced topics in linear algebra and related fields.

Gaussian Elimination

Gaussian elimination is a method used to solve systems of linear equations by performing operations on the equations to eliminate variables step by step. The goal is to transform the system into an upper triangular form (or row echelon form), from which solutions can be obtained through back substitution. This algorithmic approach simplifies complex systems and provides a systematic means of finding solutions.

Augmented Matrix

An augmented matrix is a compact representation of a system of linear equations, where the coefficients of the variables and the constants from the equations are arranged in a matrix form. This format is particularly useful when applying methods like Gaussian elimination, as it organizes all the necessary data in a structured way and facilitates systematic row operations to reach a solution.

Transcript

00:02 In order to solve this system of equations, i will be using the elimination method.

00:07 Now, since this system has three equations in it, i'm going to have to kind of break them apart and use the elimination method a few times in order to solve.

00:18 So to start, i'm just going to look at the first equation and the middle equation.

00:24 Okay, i want the zs to go ahead and cancel out because i want to have opposite coefficients.

00:31 And both of these zs already have a coefficient of two.

00:36 I just need one of them to be negative.

00:39 So i'm just going to go ahead and choose to multiply the middle equation by a negative 1.

00:44 That way when i go ahead to eliminate, my zs will cancel out.

00:49 So i'm going to rewrite this off to the side.

00:52 My top equation stays the same.

00:58 And then the middle equation, i'm multiplying everything by a negative 1.

01:09 Okay.

01:09 Now i can go ahead and use the elimination method, meaning i'm going to add the two equations together.

01:16 And i already mentioned my zs will cancel out because they have opposite coefficients.

01:24 Next, go ahead and add up all of your like terms.

01:28 So this gives me negative 4x plus y equals 7.

01:36 Okay.

01:37 I'm going to pause there with those two equations and work on the second and the second and the third equation.

01:45 Again, i want the zs to eliminate.

01:49 So here, now i'm looking at 2z and 4z...

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