Use matrices to solve the system. { 2 x-y+4 z =8 -3 x+y-2 z =5 .

Name: Problem 16, Chapter 8: Precalculus: Functions and Graphs
Uploaded: 2021-07-14T02:46:20-08:00
Duration: 5 min 29 s
Channel: Andrija Isakov
Description: Use matrices to solve the system. { 2 x-y+4 z =8 -3 x+y-2 z =5 .

step 1 This question is about linear systems in more than two variables. We want to solve this system o

Use matrices to solve the system. { 2 x-y+4 z =8 -3...

Key Concepts

Solution Set and Parametrization

Once the matrix is reduced, if the system is underdetermined or has free variables, the solution is typically presented in a parametrized form. This involves describing the solution set as a family of solutions in which one or more variables are set as parameters.

Underdetermined Systems and Free Variables

An underdetermined system arises when there are more unknowns than independent equations, leading to an infinite number of solutions. In such cases, some variables become free, meaning they can take on any value, and the remaining variables are expressed in terms of these free parameters.

Systems of Linear Equations

This concept involves solving multiple equations that share common variables. In the context of matrices, the system is represented in terms of coefficients corresponding to each variable, and solving the system means finding a set of variable values that satisfy all equations simultaneously.

Matrix Representation and Augmented Matrices

Representing a system of linear equations using matrices involves forming a coefficient matrix that holds the coefficients of the variables and an augmented matrix that appends the constants from the right-hand side of the equations. This method streamlines the process of applying algebraic procedures to solve the system.

Gaussian Elimination

Gaussian elimination is a systematic method for transforming the augmented matrix of a system into a simpler form, typically row echelon form or reduced row echelon form. This process uses elementary row operations to simplify the system, making it easier to identify the values of the variables.

Transcript

00:02 This question is about linear systems in more than two variables.

00:09 We want to solve this system of equations using matrices.

00:18 So of course to solve a linear system using matrices, we need an actual matrix for the system.

00:26 And so this is the system represented as matrix.

00:30 All the coefficients are written down in a rectangular array, where the rows correspond to equations and columns correspond to variables.

00:47 Now, since we have three equations, or sorry, three variables with two equations, we can't have just one unique solution for x, y, and z.

01:00 That's just not possible.

01:04 However, we can treat one of the variables as known, and as such, we can assign it a parameter, say t, which represents any real number, and then write the other two equations, or sorry, variables, in terms of this variable t, which is really in terms of the variable y that we started with.

01:35 Because we have two equations, we can still show some of the nature of the solution.

01:42 We can't have any, we know that we can't, or that the solution will be somewhat constrained.

01:51 And x and z, for example, would have to be specific linear combinations of one.

02:01 And so in order to find those, we can use these two equations.

02:07 On with the show.

02:10 So using matrices, we have to transform the rows...

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