Solve the system of equations by finding the reduced row echelon form for the augmented matrix.

Solve the system of equations by finding the reduced row...

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 objective is to find values for these variables that satisfy all equations simultaneously. This concept is a fundamental part of linear algebra and is essential for modeling and solving problems in various fields, such as engineering, physics, and economics.

Augmented Matrix

An augmented matrix is a compact way of representing a system of linear equations by combining the coefficient matrix with the constants from the right-hand side of the equations. This formulation simplifies the process of applying algebraic techniques, as it allows the use of matrix operations and systematic procedures to find the solution of the system.

Elementary Row Operations

Elementary row operations are basic manipulations applied to the rows of a matrix, including swapping rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another. These operations are fundamental tools in linear algebra for transforming a matrix into a simpler form without changing the system’s solution set, thereby facilitating the solving process.

Reduced Row Echelon Form (RREF)

Reduced Row Echelon Form is a specific form of a matrix achieved through the application of elementary row operations. In RREF, each leading entry in a row is 1, each leading 1 is the only nonzero entry in its column, and rows of zeros (if any) are at the bottom of the matrix. This form is particularly useful as it provides a clear and systematic method to identify the solutions of the system of linear equations, if they exist.

Transcript

00:04 We have a system of equations to solve, and what we're going to do is find the row echelon form for this system and use that to solve it.

00:13 And just keep in mind that row echelon form is not unique, so there could be multiple row echelon forms for the same system.

00:20 They should have the same solution.

00:23 And row echelon form looks like this.

00:26 That's what we're going for when possible, where the first row starts with a one, and the second row starts with a zero, then a one, and the third row.

00:34 Has a zero than a zero than a one.

00:36 So the first thing we want to do is change our system into an augmented matrix.

00:41 So the first row would be 1, 2, negative 1, 3.

00:45 The second row would be 3, 7, negative 3, 12.

00:49 And the third row would be negative 2, negative 4, 3, negative 5.

00:58 Next, what we want to do is change it to row echelon form by performing various row operations.

01:06 And we want the first row to start with a 1 and it already does so that's good so now let's look at the second row and we want it to start with a 0 so the way we're going to make that happen is we're going to take row 1 multiply it by negative 3 and add it to row 2 so row 1 stays the same but then row 2 becomes 0 1 0 3 and row 3 is going to for now stay the same okay now the next thing we want to do is we want row 3 to start with a 0.

01:52 So one way we can make that happen is if we multiply row 1 by 2 and add it to row 3...

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