Use Gauss-Jordan elimination to solve the system: {-x-y-z =1 4 x+5 y =0 y-3 z =0.

Name: Problem 90, Chapter 8: Precalculus
Uploaded: 2023-05-08T14:21:25-08:00
Duration: 8 min 4 s
Channel: Emily Quinn
Description: Use Gauss-Jordan elimination to solve the system: {-x-y-z =1 4 x+5 y =0 y-3 z =0.

Use Gauss-Jordan elimination to solve the system: {-x-y-z ...

Key Concepts

Reduced Row-Echelon Form

Reduced row-echelon form (RREF) is the final form achieved through Gauss-Jordan elimination. In RREF, each leading entry in a row is 1 and is the only nonzero entry in its column, and any zero rows appear at the bottom. This form makes it easier to read off the solutions for the variables directly, thereby providing a precise resolution to the system of equations.

Row Operations

Row operations involve three types of manipulations: swapping two rows, multiplying a row by a nonzero scalar, and adding or subtracting multiples of one row to or from another row. These operations are fundamental steps in methods such as Gaussian and Gauss-Jordan elimination because they preserve the solution set of the system while simplifying the equations.

Gauss-Jordan Elimination

Gauss-Jordan elimination is an algorithmic process used to solve systems of linear equations. It systematically applies elementary row operations to the augmented matrix of the system in order to transform it into reduced row-echelon form. This method uniquely identifies each variable by ensuring that each leading coefficient is 1 and is the only nonzero entry in its column, making it straightforward to interpret the solution.

System of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. Solving such systems requires finding values for the variables that satisfy all equations simultaneously. This is essential in many areas of mathematics, physics, economics, and engineering, since many real-world problems are modeled using such systems.

Augmented Matrix

An augmented matrix is a compact representation of a system of linear equations, where the coefficients of the variables and the constants are organized in a matrix format. The use of an augmented matrix simplifies the application of row operations since it visually consolidates the information needed to perform Gauss-Jordan elimination.

Transcript

00:01 This problem asks us to solve this system of linear equations using gauss -jordan elimination.

00:07 So what we're going to do first is put this information into a matrix.

00:12 So to do that, remember that you just put the coefficients of each variable into the matrix in the same order for each of the equations.

00:20 So the first one you've got negative 1x minus 1y minus 1z, right? now the 1 on the right -hand side of that equation is going to go to the right of this bar.

00:32 So we're going to say that it is augmented with and then we've got that 1 over there on the right.

00:36 We're going to do the same thing for the rest of the rows.

00:38 The next one you've got 4x plus 5y, but you don't have any z's.

00:44 So we're going to put 0 there equals 0.

00:46 And then the last one we've got no x's, but we have 1y minus 3z equals 0.

00:53 Now gauss -jordan elimination means that we're going to use elementary row operations to take this matrix and turn it into one where you've got your leading ones, you have zeros above and below the leading ones, and any all zero rows are at the bottom.

01:09 So first elementary row operation i'm going to do is scale that first row by a negative 1.

01:15 So i'm going to say negative row 1 and we're going to do that so that we've got a first entry in that row that's a positive 1.

01:23 So that gives me positive 1, positive 1, positive 1, negative 1, right? because those operations go all the way across even to the numbers to the right of that bar.

01:34 So now we're here.

01:35 Now i'm going to use that positive 1 here, which is our leading 1 or sometimes called the pivot.

01:41 And we're going to use that to zero out this entry.

01:44 So i need row 2 to change.

01:46 So i'm going to take row 2 minus 4 row 1 because subtracting 4 times this top entry is going to give us a 0 there.

01:56 Okay, but again, you have to do it all the way through.

01:59 So my first row will stay the same 1, 1, 1, negative 1.

02:03 The second row when i do the subtracting this entry minus 4 times this entry gets us 0 here.

02:12 Then you've got 5 minus 4 times this entry.

02:17 So that's 5 minus 4, which is 1.

02:20 And then here you have 0 minus 4 times 1, which would be negative 4.

02:26 Do the same thing to the right.

02:29 So here we have 0 minus 4 times negative 1.

02:33 So that's plus 4, so positive 4.

02:36 And then the bottom row for the moment stays the same.

02:39 Now in that first row, i have my leading 1.

02:43 In my second row, i have my leading 1.

02:46 That's the first non -zero entry in that particular row.

02:50 I need to use that entry, this is our new pivot, to make the entries both above and below it 0.

02:58 So to do that, i'm going to take row 1 minus row 2 because that means i'm going to be doing 1 minus 1, which will make that upper 1 turn to 0.

03:09 So you have 1 minus 0, which is 1.

03:15 1 minus 1 is 0.

03:19 1 minus a negative 4.

03:21 Watch your answers there carefully.

03:22 Watch your negatives.

03:23 That'd be a 5.

03:24 And then negative 1 minus 4 would be negative 5.

03:30 Okay? last two rows stay the same.

03:34 Okay, we only do one operation at a time.

03:36 That's why they're called elementary row operations because they're supposed to be not necessarily easy to do, but they're simple sort of one -step operations.

03:45 Okay? next, we're going to take this same pivot and we're going to use it to zero out the entry below it.

03:51 So i need to make this entry a 0.

03:54 Well, to do that, i'm going to take row 3 minus row 2 since, again, those are opposites, or need to be opposites, to be able to make that 0 out.

04:04 Now, the first row here is not changing, so that's going to stay the same.

04:09 The second row also isn't changing, but the third one is...

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