In Problems 17-54, solve each system of equations. If the system has no solution, say that it is

In Problems 17-54, solve each system of equations. If the...

Key Concepts

System of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. The goal is to find values for these variables that satisfy all equations simultaneously. Such systems can have a unique solution, an infinite number of solutions, or no solution at all, and they are a fundamental topic in algebra and linear algebra.

Elimination Method

The elimination method involves strategically adding or subtracting equations from one another to eliminate one or more variables. This reduction simplifies the system into one with fewer variables, making it easier to solve step by step until the values of all unknowns are determined. It is particularly useful in systems with multiple equations.

Gaussian Elimination

Gaussian elimination is a systematic application of the elimination method where the system of linear equations is represented as an augmented matrix. By performing row operations on this matrix to achieve an upper triangular or row-echelon form, the system becomes simpler to solve through back-substitution. This method is essential in both theoretical and computational aspects of linear algebra.

Consistency and Inconsistency in Systems

Consistency in a system of linear equations means that there is at least one set of values for the variables that satisfies all the equations simultaneously. In contrast, an inconsistent system has no solution because the equations contradict each other. Determining whether a system is consistent or inconsistent is one of the key steps in analyzing systems of equations.

Transcript

00:01 The concept involved with this problem is to solve a system of three equations and three variables using any method that you wish.

00:12 I am going to solve this system by using the method of elimination.

00:19 There are other methods you have studied, but this one to me looked like a good one to use elimination loop.

00:27 So what i'm going to do with this, when you use elimination, you've got to pick two equations.

00:33 And eliminate one of the variables.

00:36 So on this one, i'm going to take equations 1 and 2 and eliminate the y.

00:46 Okay, so this is equation 1 and equation 2, and we need to eliminate the y.

01:07 So what i'm going to do to do that, i'm going to take the second equation and multiply it by negative.

01:17 2.

01:19 Okay, so the first equation is going to say what it is and i'm going to multiply that second equation by negative 2 and then add those two equations together.

01:42 I lose my ys and i get an equation with just an x and a z.

01:50 Okay, now i need to pick another pair of equations and also eliminate the y.

01:56 So i'm going to pick this time equations 1 and 3 and then we have to eliminate the to help ourselves okay so here's equation one again and then equation three now in order to eliminate the y i'm going to turn this 2y in the second equation into a negative 2 y by multiplying by a negative 1 okay so i'm going to keep my first equation and i'll distribute that negative 1 through the second equation whoops 14 14 thing getting ahead of myself...

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