How Form 8865 Reduction 2023-2024

In this video, we're gonna focus on using Gaussian elimination to solve a system of equations with three variables. So, let's start with this problem. Let's say we have X + Y - Z = -2, 2X - Y + Z = 5, and -X + 2Y + 2Z = 1. So, how can we use matrices to solve this system of equations? The first thing we need to do is convert it to an augmented matrix. Let's write the coefficients in this matrix. We have [1, 1, -1 | -2], [2, -1, 1 | 5], and [-1, 2, 2 | 1]. Now, I'm gonna convert this matrix into row-echelon form. I want these three numbers to be 1, and I want these numbers to be 0. So, let's make those 3 numbers in blue first. To do this, I'm gonna add Row 1 and Row 3 together and apply that change to Row 3. So, let's rewrite the matrix. The first row is not going to change, all of the changes will be applied to Row 3. Next, I want to convert the 2 into a 0. I'm going to apply the changes to Row 2. I'm going to multiply Row 1 by -2 and add that to Row 2. Now, what do you think is the next thing we should do at this point? We need the number in the fourth column to be a 0. To do this, we just need to add Rows 2 and 3 together and apply the changes in Row 3. If I wanted to, I could convert this back to a system of linear equations and use back substitution to get X, Y, and Z. But what I want to do is convert it to row-echelon form. Now, I'm going to convert...