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

Systems of Linear Equations

A system of linear equations is a collection of one or more linear equations involving the same set of variables. These systems can have a unique solution, infinitely many solutions, or no solution. The process of solving these systems involves finding values for the variables that simultaneously satisfy all the given equations.

Augmented Matrix

An augmented matrix is a matrix that includes both the coefficients of the variables and the constants from the right-hand side of the equations. This form is particularly useful for applying row operations directly to the entire system in order to simplify or solve it.

Row Operations

Row operations are the procedures used to manipulate the rows of a matrix. These operations include swapping rows, multiplying a row by a nonzero scalar, and adding or subtracting a multiple of one row to another. They are essential tools in Gaussian elimination to simplify the augmented matrix to a form where the solution to the system can be easily read.

Gaussian Elimination

Gaussian elimination is a systematic method of performing row operations on an augmented matrix to bring it into row-echelon form or reduced row-echelon form. This method simplifies the process of solving a system of linear equations by making it easier to perform back substitution or directly read off the solutions.

Consistency and Inconsistency

Consistency in a system of equations indicates that there is at least one set of values for the variables that satisfies all the equations, while inconsistency means that no such set exists. In the context of matrices and row operations, an inconsistent system is typically revealed by a row in the augmented matrix that represents an impossible equation, such as 0 = nonzero.

Matrices

A matrix is a rectangular array of numbers which can represent coefficients in a system of linear equations. Matrices help to compactly express and solve these systems by organizing the coefficients into rows and columns, especially when using methods like row operations or Gaussian elimination.

Transcript

00:01 In this problem, our objective is to solve a system of linear equations using matrices.

00:08 Using matrices to solve a system of linear equations will require that we go through a series of row operations on the matrix.

00:17 Now as you go through the row operations, your goal is to try to transform your matrix into a reduced row echelon form.

00:26 This is the reduced row echelon form for a system that has three equations and three variables.

00:34 Now let's assume the variables are x, y, and z in that order.

00:40 If you have the reduced row echelon form, then you should be able to conclude that from your system, x has a value of some real number a, that y would equal some real number b, and that z would equal some rule number c, meaning the order triple a, b, c, would be the solution to your system.

01:07 Now, as i work through this problem, a couple notations that i'm going to use.

01:12 The lowercase r1, r2, r3, lower case r sub 1, r sub 2, r sub 3 will refer to the rows of the current matrix that i'm working with.

01:27 And the uppercase r sub 1, r so 2, r7, 3 will refer to the row operation that i will perform.

01:39 Okay, so with that in mind, let's start looking through this problem, solving the system with matrices.

01:46 Okay, you were given a system that has three equations, three variables.

01:51 So i'm going to set up the augmented matrix for that system.

01:55 Equation 1, using the coefficients of x, y, and z, you have this row, along with the constant of one.

02:06 Okay, in the same way, equation two, you would have this row.

02:13 Equation 3 would give you this row.

02:19 Okay, so we're going to use this matrix, and with row operations, we're going to try to put it in the reduced row atchalant form.

02:28 Now i'm going to go through a series of row operations.

02:31 I'll tell you what i'm doing on each one of the steps.

02:34 Keep in mind that what we're trying to do is to produce these ones and zeros that are on the left hand side of the vertical bar.

02:42 There's not any one right way to do the problem.

02:45 You're just working it with trying to take what hits you and produce your ones and you zeros.

02:52 So when i look at the matrix i'm given, i'm going to make a change to row two.

02:58 And the operation that i'm going to do that is to take row one times a negative two.

03:05 I'm so row one i'll fix what that looks like row one times a negative two and add that to row two in the same step i want to produce a new row three by taking row one times a negative three and add that to row three and you can do more than one step more than one row operation in each step so since i'm not doing any change to row one i'm going to go ahead and copy it down the rows that i'm not changing, i like to copy them down first and then make my changes.

03:45 Okay, so i'm going to start off and change row 2 by taking row 1 times negative 2 and add that answer to row 2.

03:53 And that's going to produce this row 2.

04:03 And i'm going to change row 3 by again taking row 1, but this time multiplied by negative 3 and adding that to row 3.

04:12 And this is what i will produce.

04:19 Interesting.

04:20 Notice i did enter in a lot of the zeros that we need.

04:24 Let's see what happens from here.

04:29 On my next row operation, i'm going to take row two, and i'm going to produce a new row two, i'm multiplying it by a negative one -fifth.

04:43 And i'm going to do the same type of thing to produce a new row three.

04:52 Okay.

04:53 Again, i'm not changing row one, keeping it.

05:00 For row two, when i multiply each one of these by negative one and fifth.