Solve each system of equations using matrices (row operations). If the system has no solution, s

Solve each system of equations using matrices (row...

Key Concepts

System of Linear Equations

A system of linear equations consists of multiple linear equations that share common variables. The solution to the system is found by identifying values for the variables that satisfy all the equations simultaneously. Such systems can have a single solution, infinitely many solutions, or no solution at all.

Augmented Matrix

An augmented matrix is a structured representation of a system of linear equations. It includes both the coefficients of the variables and the constants from the right-hand side of the equations in a single matrix format. This representation is especially useful in applying systematic techniques like row operations to solve the system.

Elementary Row Operations

Elementary row operations include interchanging rows, multiplying a row by a nonzero constant, and adding a multiple of one row to another row. These operations are used to simplify the augmented matrix to a form that makes it easier to directly read off the solutions of the system.

Gaussian Elimination

Gaussian elimination is a method used to solve systems of linear equations by systematically applying elementary row operations to transform the augmented matrix into an upper triangular or row echelon form. This process simplifies the system, allowing for easier determination of the solution through back substitution.

Inconsistency and Consistency in Systems

A system is consistent if there is at least one set of values for the variables that satisfies all the equations simultaneously; it is inconsistent if no such set exists. When performing row operations, a row that results in all zeros for the coefficients but has a non-zero constant indicates the system is inconsistent.

Transcript

00:01 The idea or the concept behind this problem is to solve a system of linear equations using matrices.

00:10 Along with this, we will work with several row operations and try to take the matrix that we write from the system and transform the matrix into this reduced row echelon form.

00:27 Now, if you can get a matrix from a system into a system, to the reduced row echelon form, you will be able to determine the solution to the system.

00:38 Now from the reduced row echelon form, the first line tells you that the x variable of your system is going to equal sum wheel number a, and that the y variable will equal some wheel number b, and the z variable will equal some wheel number c, which then makes the order triple for the system abc.

01:03 So again, our goal is to take the matrix from the system, perform some row operations on that matrix, and transform the matrix into this reduced row echelon form.

01:17 Now, as i work through this problem, i'm going to let lowercase r sub 1, r sub 2, and r sub 3 stand for the row of the current matrix that i am working with.

01:37 And i'm going to let uppercase r sub 1, r sub 2, r73 represent the row operation that i will be performing.

01:51 Okay, so with all that in mind, let's start this problem.

01:55 I'm given a system.

01:57 My first step will be to write the matrix that is represented by this system.

02:04 Now the first row of my matrix will come from the first equation in the system.

02:10 I'll start off with the coefficients of the variables, vertical bar, and then the constant.

02:18 Okay, second row will be from the second equation, and the third row will be from the third equation.

02:28 Now i would suggest that you always double check and make sure that you have not made any mistake.

02:33 It's so easy to leave off a negative sign of wherever, but double check and make sure you're always double check and make sure you...

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