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

Name: Problem 44, Chapter 8: College Algebra
Uploaded: 2021-07-21T01:20:26-08:00
Duration: 8 min 2 s
Channel: Stephanie Carter
Description: Solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent. { (1)/(2) x+y= -2 x-2 y= 8 .

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

Key Concepts

Systems of Linear Equations

Systems of linear equations consist of multiple linear equations that share common variables. The objective is to find the set of values which satisfy all equations simultaneously. These systems are foundational in linear algebra and appear in various applications such as engineering, economics, and the natural sciences.

Matrices and Augmented Matrices

A matrix is an organized array of numbers that can represent the coefficients of a system of linear equations. When an additional column containing the constants from the equations is added, the result is an augmented matrix. This provides a compact and efficient way to represent and manipulate systems of equations using algebraic methods.

Row Operations

Row operations are elementary manipulations performed on the rows of a matrix, including row swapping, scaling a row by a nonzero constant, and adding or subtracting multiples of one row from another. These operations preserve the solution set of the corresponding system of equations and are fundamental to methods like Gaussian elimination.

Reduced Row Echelon Form

The reduced row echelon form (RREF) is a standardized form of a matrix achieved through a sequence of row operations. In the RREF, each leading entry is 1, and it is the only nonzero entry in its column. This form simplifies the process of solving systems, as it allows for straightforward back-substitution to determine the variable values.

Inconsistent Systems

An inconsistent system is one that has no solution because the equations contradict each other. In the context of matrices, inconsistency is often revealed when a row in the augmented matrix reduces to an equivalent of a false statement, such as 0 = 1. This indicates that the given system cannot be satisfied by any assignment of variables.

Transcript

00:01 In this problem, we will use the concept of matrices and row operations to solve a system of equations.

00:10 Our goal with the matrices and the row operations is to go through some steps to change or transform our matrix into what is called a reduced row echelon form matrix.

00:26 The matrix you see in blue here is a reduced row echelon form matrix.

00:31 Form.

00:33 Notice on this the first row is 1, 0, and the a stands for some real number, any real number that works with that problem.

00:43 Row 2 is 0, 1, and some real number b.

00:48 Now if you have a matrix in this form, then you would be able to conclude that 1x would equal a and that 1y would equal b.

01:03 Meaning that the solution to the system you're working with would be the ordered pair a -b.

01:10 So that's your goal is you want to take the matrix you can form from the given system.

01:17 Use row operations to transform that matrix into a reduced row etchlon form matrix, but you can then get the solution for your system.

01:32 Okay, so as i'm working through the problems, you will see me, a lowercase r sub 1 and a lower case r sub 2.

01:40 And what those notations indicate is that those are rows of the current matrix.

01:56 And when you see the uppercase r sub 1, upper case r sub 2, those are going to involve rows that i will perform operations on to get a new matrix.

02:14 Alright, so let's get started on this problem.

02:18 I have this system given that i'm going to try to solve with matrices, so i'm going to form a matrix.

02:26 And remember the first column or the coefficients of your x variable, second column is the coefficients of your y variables, you will have a vertical bar, and then you will have your constants on the right hand side of that bar.

02:44 So, for the first row, using the first equation, i will have one half, one, and negative 2.

02:55 The second row i will get from my second equation, 1, negative 2, and positive 8.

03:04 Now i'm going to use row operations on this matrix that i have in green and go through a series of steps to transform it into a reduced row echelon.

03:19 Alright, so the first operation that i'm going to do, and there's different ways to approach these problems, but the first operation i'm going to do is i'm going to transform my row 1.

03:31 I really don't like that equation, i mean that fraction there is the coefficient of x.

03:35 So i'm going to take row 1 and i'm going to multiply it times 2.

03:40 Take the current row 1 times 2 and i will produce this matrix 1, 2, 2, negative 4.

03:53 I'm just going to copy over row 2.

03:55 I'm not going to change it in this step.

04:01 Okay, then i'm going to make another new row 1 and change colors.

04:09 Okay, we'll use that...

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