Gaussian Elimination & Row Echelon Form

1. Formula
   Observation

   The variable x is written in the first equation and appears as the first column of the augmented matrix.

x

First unknown variable in the system of three linear equations.

Real-valued unknown; no explicit domain stated in the video.

1. Formula
   Observation

   The variable y is written in the equations and corresponds to the second column of the augmented matrix.

y

Second unknown variable in the system of three linear equations.

Real-valued unknown; no explicit domain stated in the video.

1. Formula
   Observation

   The variable z is written in the equations and corresponds to the third column of the augmented matrix.

z

Third unknown variable in the system of three linear equations.

Real-valued unknown; no explicit domain stated in the video.

1. Formula
   Observation

   A bracketed array with a vertical bar is drawn to the right of the system.

2. Audio
   Observation

   The narrator says to convert the system into an augmented matrix and use a vertical bar to separate the left side from the right side.

[A | b]

Augmented matrix formed by placing the coefficients of x, y, z on the left of a vertical bar and the constants on the right.

Matrix representation of a linear system.

1. Formula
   Observation

   The notation $R_1 + R_3$ is written above the arrow leading to the next matrix.

2. Audio
   Observation

   The narrator says he will add row one and row three together and apply that change to row three.

$R_1 + R_3 \to R_3$

Elementary row operation replacing row 3 by the sum of row 1 and row 3.

Row operation on the augmented matrix.

1. Formula
   Observation

   The augmented matrix shown at the start is [[1,1,-1|-2],[2,-1,1|5],[0,3,1|-1]].

[[1,1,-1|-2],[2,-1,1|5],[0,3,1|-1]]

Augmented coefficient matrix for a three-variable linear system before the next row reduction step.

Entries are real numbers; rows correspond to equations and columns correspond to variables plus constants.

1. Formula
   Observation

   The arrow label above the first transformation is -2R_1 + $R_2$.

2. Audio
   Observation

   The speaker says, 'It's called a matrix row operation.'

-2R_1+$R_2$

Replace row 2 by the sum of -2 times row 1 and row 2.

Applies to the current augmented matrix; only row 2 changes.

1. Formula
   Observation

   The second arrow label is $R_2+R_3$.

2. Audio
   Observation

   The speaker says, 'So it's going to be R two plus R three.'

$R_2+R_3$

Replace row 3 by the sum of row 2 and row 3.

Applies to the intermediate matrix after the first row operation; only row 3 changes.

1. Formula
   Observation

   $R_1$ appears in the row-operation labels and in the spoken explanation of entries.

$R_1$

First row of the current augmented matrix.

Used as a source row in elementary row operations.

1. Formula
   Observation

   $R_2$ appears in both row-operation labels and in the spoken entry-by-entry calculations.

$R_2$

Second row of the current augmented matrix.

First used as the target row in -2R_1+$R_2$, then as a source row in $R_2+R_3$.

1. Formula
   Observation

   $R_3$ appears in the second row-operation label and in the spoken calculation of the new third row.

$R_3$

Third row of the current augmented matrix.

Target row in the second elementary row operation.

1. Diagram
   Observation

   $A 3\times 4$ augmented matrix is written on the left at the start and later transformed into $a 3\times 4$ upper-triangular augmented matrix on the right.

[ [1, 1, -1 | -2], [0, -3, 3 | 9], [0, 0, 4 | 8] ]

Augmented coefficient matrix for a three-variable linear system before normalization to row echelon form.

Entries are real numbers; the vertical bar separates coefficient columns from the constants column.

1. Audio
   Observation

   The narrator introduces a system of equations with three variables.

2. Formula
   Observation

   Three equations are written: $x + y - z = -2$, $2x - y + z = 5$, and -x + $2y + 2z = 1$.

The video begins with a concrete linear system containing three unknowns x, y, and z. Each equation is linear, with coefficients multiplying the variables and constants on the right-hand side.

1. The system has three equations and three variables.

2. The video does not state a domain for the variables.

1. Audio
   Observation

   The narrator says the first thing to do is convert the system to an augmented matrix.

2. Formula
   Observation

   The matrix entries are filled row by row from the coefficients and constants of the three equations.

Each row of the augmented matrix corresponds to one equation. The entries before the vertical bar are the coefficients of x, y, and z in order; the entry after the vertical bar is the constant term from the right-hand side of that equation.

1. The variables must be placed in a consistent order across all equations.

2. Missing variables would require zero coefficients, but this example has all three variables present in every equation.

1. System of three linear equations in three variables

1. Audio
   Observation

   The narrator says he wants to convert the matrix into row echelon form.

2. Diagram
   Observation

   Red circles mark the diagonal positions intended to become 1, and blue circles mark positions below the first pivot intended to become 0.

1. The video visually indicates the target pattern but does not give a full formal definition of row echelon form.

In the demonstrated setup, the goal is to make the leading diagonal entries equal to 1 and the entries below the first pivot equal to 0. The visual marking identifies which positions are being targeted before the row operation is performed.

1. This is the specific target shown for the current 3-by-3 augmented matrix.

2. The video does not explicitly state all general row-echelon-form conditions.

1. Constructing the augmented matrix from a linear system

1. Audio
   Observation

   The narrator says he will add row one and row three together and apply that change to row three.

2. Formula
   Observation

   The operation is written as $R_1 + R_3$ above the transformation arrow.

The operation replaces row 3 with the entrywise sum of row 1 and row 3. Rows 1 and 2 are copied unchanged, while each entry in row 3 is computed by adding the corresponding entries from rows 1 and 3.

1. The operation is applied to the augmented matrix.

2. The vertical bar is preserved, so the constant column is transformed together with the coefficient columns.

1. Constructing the augmented matrix from a linear system
2. Target pattern for row echelon form in this example

1. Audio
   Observation

   The speaker names the process a matrix row operation and explains that one row is multiplied by -2 and added to another row.

2. Formula
   Observation

   The written operation is -2R_1+$R_2$.

The video demonstrates replacing row 2 with -2 times row 1 plus row 2 so that the first-column entry in row 2 becomes 0. The method is applied column by column while rows 1 and 3 remain unchanged.

1. Applied to an augmented matrix.

2. Only the target row changes.

3. The multiplier is chosen to cancel the pivot-column entry in the target row.

1. Audio
   Observation

   The speaker says, 'Now, we need this number to be a zero,' referring to the (3,2) entry, then applies $R_2+R_3$.

2. Formula
   Observation

   The second operation is written as $R_2+R_3$ and produces a third row [0,0,4|8].

After the first elimination, the matrix has a nonzero entry 3 below the second pivot position. The video adds row 2 to row 3 to make the (3,2) entry zero, advancing the matrix toward upper-triangular form.

1. Applied to the intermediate matrix after the first row operation.

2. Only row 3 changes.

3. The goal is to zero the entry below the second pivot position.

1. Elementary row operation used to eliminate a leading entry

1. Audio
   Observation

   The speaker says he wants to convert the matrix to row echelon form and turn the leading diagonal entries into ones.

2. Diagram
   Observation

   Circles mark the diagonal entries 1, -3, and 4 in the starting matrix.

The video treats row echelon form as the target shape obtained by making the leading nonzero entry in each pivot row equal to 1. The first pivot is already 1, so only rows 2 and 3 need scaling.

1. The matrix is already in upper-triangular shape with zeros below the main diagonal.

2. The intended next step is normalization of pivots to 1.

1. Formula
   Observation

   The arrow label shows -1/3 $R_2$.

2. Audio
   Observation

   The speaker explains multiplying row two by negative one third, equivalently dividing by negative three.

3. Diagram
   Observation

   The resulting second row is written as 0, 1, -1 | -3.

To normalize the second pivot, every entry in row 2 is multiplied by -1/3. This changes -3 to 1, 3 to -1, and 9 to -3 while leaving the leading zero unchanged.

1. Applied to the whole second row of the augmented matrix.

2. Used after the matrix is already upper triangular.

1. Goal of converting to row echelon form

1. Formula
   Observation

   The arrow label shows $1/4 R_3$.

2. Audio
   Observation

   The speaker says row three is multiplied by one fourth.

3. Diagram
   Observation

   The resulting third row is written as 0, 0, 1 | 2.

To normalize the third pivot, every entry in row 3 is multiplied by $1/4$. This changes 4 to 1 and 8 to 2 while preserving the zeros in the first two positions.

1. Applied to the whole third row of the augmented matrix.

2. Used after the matrix is already upper triangular.

1. Goal of converting to row echelon form

1. Audio
   Observation

   The speaker says to plug z into the second equation to get y, then plug y and z into the first equation to get x.

2. Diagram
   Observation

   Equations are rewritten sequentially from bottom to top until $x = 1$ is boxed.

Once the augmented matrix is converted back into equations, the bottom equation gives z directly. That value is substituted upward into the second equation to solve for y, and then both y and z are substituted into the first equation to solve for x.

1. The system has been reduced to an upper-triangular augmented matrix with unit pivots.

2. Variables are read from left to right as x, y, z.

1. Goal of converting to row echelon form
2. Matrix columns correspond to variables

1. Diagram
   Observation

   Red x, y, z labels are placed above the first three matrix columns before the rows are rewritten as equations.

2. Audio
   Observation

   The speaker corrects an initial $x = 2$ reading to $z = 2$ by matching the last row to the z column.

Each coefficient column keeps the same variable identity: the first column is x, the second is y, and the third is z. Reading a row as an equation requires this column-to-variable mapping.

1. The matrix columns retain the order x, y, z.

1. Audio
   Observation

   Speaker says, 'this form is row echelon form where you have a diagonal of ones and zeros beneath it.'

2. Diagram
   Observation

   The right-hand matrix shown is [[1,1,-1|-2],[0,1,-1|-3],[0,0,1|2]], with leading 1s on the diagonal and zeros below.

In this clip, row echelon form is described as a matrix shape with a diagonal of 1s and zeros beneath that diagonal. The speaker also states that the entries above the diagonal do not matter for this description.

1. Applies to the coefficient part of the augmented matrix shown.

2. The video emphasizes leading 1s on the diagonal and zeros below them.

1. Audio
   Observation

   The speaker says, 'It's called a matrix row operation.'

The transformation being applied to the augmented matrix is called a matrix row operation.

1. The speaker is referring to the written operation -2R_1+$R_2$.

No explicit quantifier beyond the demonstrated operation.

1. Audio
   Observation

   The speaker says, 'Now, we need this number to be a zero. So all we got to do is add rows two and three.'

2. Formula
   Observation

   The circled target entry is the 3 in position (3,2), and the next written operation is $R_2+R_3$.

To make the (3,2) entry zero, the video adds row 2 to row 3 and records the result in row 3.

1. The current matrix is [[1,1,-1|-2],[0,-3,3|9],[0,3,1|-1]].

For the displayed matrix at this stage.

1. Audio
   Observation

   The speaker says he wants to convert these three numbers into a one and notes the first is already one.

2. Diagram
   Observation

   The diagonal entries 1, -3, 4 are circled.

In the worked example, the entries on the main diagonal of the upper-triangular augmented matrix are made equal to 1 by scaling the corresponding rows.

1. The matrix is already upper triangular.

2. Each displayed pivot is nonzero.

For each pivot row shown in the example.

1. Audio
   Observation

   The speaker says column one is the coefficients for x, this is for y, and z.

2. Diagram
   Observation

   Red labels x, y, z are placed above the first three columns.

In the augmented matrix, the first, second, and third coefficient columns correspond respectively to the unknowns x, y, and z.

1. The augmented matrix represents a three-variable linear system.

2. Columns are ordered consistently from left to right.

For the three coefficient columns in this example.

1. Audio
   Observation

   'this form is row echelon form where you have a diagonal of ones and zeros beneath it. It really doesn't matter what's here.'

2. Diagram
   Observation

   Blue marks highlight the diagonal 1s and the zero region below them in [[1,1,-1|-2],[0,1,-1|-3],[0,0,1|2]].

1. The statement is given informally rather than as a full formal definition with all standard row-echelon conditions.

For the displayed $3\times 3$ coefficient block, row echelon form is characterized by having a diagonal of 1s and zeros beneath that diagonal, while the entries above the diagonal are irrelevant to this description.

1. The matrix is in the displayed upper-triangular pattern with leading 1s on the diagonal.

Stated for the example matrix shown on screen.

1. Formula
   Observation

   The board lists -3(2)+$2(3)=0$, -3(1)+$2(2)=+1$, -3(-1)+$2(1)=5$, and -3(1)+$2(10)=17$.

2. Diagram
   Observation

   The transformed matrix on the right has second row 0, 1, 5, 17.

Applying -3R_1+$2R_2$ to $\left[\begin{array}{ccc|c}2&1&-1&1\\3&2&1&10\\0&2&-3&-5\end{array}\right]$ gives $\left[\begin{array}{ccc|c}2&1&-1&1\\0&1&5&17\\0&2&-3&-5\end{array}\right]$.

1. The starting matrix is the one shown on the board.

2. The operation replaces row 2 only.

For each column $j=1$,2,3,4, the new (2,j) entry equals -3 times the old (1,j) entry plus 2 times the old (2,j) entry.

1. Audio
   Observation

   The speaker says for row 2 and 3 column 1 it is 0 and 0, so negative 2 times 0 plus 0 is 0.

2. Diagram
   Observation

   The third matrix begins with first column 2, 0, 0.

In the intermediate matrix, applying -2R_2+$R_3$ leaves the first entry of row 3 equal to 0 because -2\cdot $0+0=0$.

1. The current matrix is the one after the first row operation.

2. The operation targets row 3.

Specifically for column 1 of row 3.

1. Audio
   Observation

   The speaker says, "So negative two plus two, that will give us the zero that we wanted."

2. Formula
   Observation

   The computation -2(1)+$2=0$ is written on screen.

For the displayed matrix, applying -2R_2+$R_3$ makes the (3,2) entry equal to 0 because -2(1)+$2=0$.

1. The current (2,2) entry is 1.

2. The current (3,2) entry is 2.

3. The operation replaces row 3 by -2 times row 2 plus row 3.

For the specific matrix shown in this example.

1. Audio
   Observation

   The speaker says, "Now, once you have these three zeros, you can go ahead and get all the answers without putting it in row echelon form, which is what we're going to do in this example."

1. The speaker's wording contrasts this stage with further reduction to row echelon form; the clip does not define row echelon form formally.

After obtaining the three zeros below the pivots, the system can be solved directly by translating the triangular matrix into equations and using back substitution.

1. The augmented matrix has been reduced to upper-triangular form.

2. The coefficient columns correspond to variables x, y, z.

For the worked example shown in the clip.

1. Audio
   Observation

   The narrator computes each entry of the new third row aloud: 1 plus negative 1, 1 plus 2, negative 1 plus 2, and negative 2 plus 1.

2. Formula
   Observation

   The resulting matrix is written with third row 0, 3, 1, -1.

1. Expression

   $$\left[\begin{array}{ccc|c} 1 & 1 & -1 & -2 \\ 2 & -1 & 1 & 5 \\ -1 & 2 & 2 & 1 \end{array}\right]$$
   Explanation

   Start from the augmented matrix built directly from the system.

   Justification

   Definition of the augmented matrix from the given equations.

   Shown in the video
2. Expression

   $$R_3 \leftarrow R_1 + R_3$$
   Explanation

   Apply the elementary row operation that replaces row 3 by the sum of row 1 and row 3.

   Justification

   Stated by the narrator and written above the transformation arrow.

   Shown in the video
3. Expression

   $$1 + (-1) = 0$$
   Explanation

   Compute the first entry of the new row 3.

   Justification

   Entrywise addition of the first entries of rows 1 and 3.

   Shown in the video
4. Expression

   $$1 + 2 = 3$$
   Explanation

   Compute the second entry of the new row 3.

   Justification

   Entrywise addition of the second entries of rows 1 and 3.

   Shown in the video
5. Expression

   $$-1 + 2 = 1$$
   Explanation

   Compute the third entry of the new row 3.

   Justification

   Entrywise addition of the third entries of rows 1 and 3.

   Shown in the video
6. Expression

   $$-2 + 1 = -1$$
   Explanation

   Compute the augmented constant entry of the new row 3.

   Justification

   Entrywise addition of the right-side entries of rows 1 and 3.

   Shown in the video
7. Expression

   $$\left[\begin{array}{ccc|c} 1 & 1 & -1 & -2 \\ 2 & -1 & 1 & 5 \\ 0 & 3 & 1 & -1 \end{array}\right]$$
   Explanation

   Write the transformed augmented matrix with rows 1 and 2 unchanged and row 3 replaced by the computed sum.

   Justification

   Result of the elementary row operation.

   Shown in the video

After applying $R_1 + R_3 \to R_3$, the augmented matrix becomes $\left[\begin{array}{ccc|c} 1 & 1 & -1 & -2 \\ 2 & -1 & 1 & 5 \\ 0 & 3 & 1 & -1 \end{array}\right]$.

1. Audio
   Observation

   The speaker computes each entry of the new row 2 aloud, column by column.

2. Formula
   Observation

   The written calculations include -2(1)+$2=0$, -2(1)+(-1)=-3, -2(-1)+$1=3$, and -2(-2)+$5=9$.

1. Expression

   $$R_2 \leftarrow -2R_1+R_2$$
   Explanation

   The operation replaces row 2 by -2 times row 1 plus row 2.

   Justification

   Stated verbally and written on screen as the matrix row operation.

   Shown in the video
2. Expression

   $$-2(1)+2=0$$
   Explanation

   Column 1 of the new row 2 is computed from row 1 entry 1 and row 2 entry 2.

   Justification

   Direct substitution into the written operation.

   Shown in the video
3. Expression

   $$-2(1)+(-1)=-3$$
   Explanation

   Column 2 of the new row 2 is computed from row 1 entry 1 and row 2 entry -1.

   Justification

   Direct substitution into the written operation.

   Shown in the video
4. Expression

   $$-2(-1)+1=3$$
   Explanation

   Column 3 of the new row 2 is computed from row 1 entry -1 and row 2 entry 1.

   Justification

   Direct substitution into the written operation.

   Shown in the video
5. Expression

   $$-2(-2)+5=9$$
   Explanation

   Column 4 of the new row 2 is computed from row 1 entry -2 and row 2 entry 5.

   Justification

   Direct substitution into the written operation.

   Shown in the video
6. Expression

   $$\begin{bmatrix}1&1&-1&-2\\0&-3&3&9\\0&3&1&-1\end{bmatrix}$$
   Explanation

   Rows 1 and 3 stay the same, and the newly computed row 2 is inserted.

   Justification

   The speaker says everything else will be the same and rewrites the other numbers.

   Shown in the video

Applying -2R_1+$R_2$ transforms [[1,1,-1|-2],[2,-1,1|5],[0,3,1|-1]] into [[1,1,-1|-2],[0,-3,3|9],[0,3,1|-1]].

1. Audio
   Observation

   The speaker says, 'So if we add R two plus R three from column one, that's zero plus zero... And for column two, R two plus R three is going to be negative three plus three, which is zero... And then for column three, it's three plus one... And then for the fourth column, nine plus negative one is eight.'

2. Formula
   Observation

   The written operation is $R_2+R_3$ and the resulting third row is [0,0,4|8].

1. Expression

   $$R_3 \leftarrow R_2+R_3$$
   Explanation

   The operation replaces row 3 by the sum of row 2 and row 3.

   Justification

   Written on the arrow and stated verbally.

   Shown in the video
2. Expression

   $$0+0=0$$
   Explanation

   Column 1 of the new row 3 is obtained by adding the first entries of rows 2 and 3.

   Justification

   Direct substitution from the intermediate matrix.

   Shown in the video
3. Expression

   $$-3+3=0$$
   Explanation

   Column 2 of the new row 3 is obtained by adding the second entries of rows 2 and 3.

   Justification

   Direct substitution from the intermediate matrix.

   Shown in the video
4. Expression

   $$3+1=4$$
   Explanation

   Column 3 of the new row 3 is obtained by adding the third entries of rows 2 and 3.

   Justification

   Direct substitution from the intermediate matrix.

   Shown in the video
5. Expression

   $$9+(-1)=8$$
   Explanation

   Column 4 of the new row 3 is obtained by adding the constant entries of rows 2 and 3.

   Justification

   Direct substitution from the intermediate matrix.

   Shown in the video
6. Expression

   $$\begin{bmatrix}1&1&-1&-2\\0&-3&3&9\\0&0&4&8\end{bmatrix}$$
   Explanation

   Rows 1 and 2 remain unchanged, and the computed row is written as the new row 3.

   Justification

   The speaker states that row ones and row two will stay the same, then writes the final matrix.

   Shown in the video

Applying $R_2+R_3$ transforms [[1,1,-1|-2],[0,-3,3|9],[0,3,1|-1]] into [[1,1,-1|-2],[0,-3,3|9],[0,0,4|8]].

1. Formula
   Observation

   Two stacked operation labels appear between the matrices: -1/3 $R_2$ and $1/4 R_3$.

2. Diagram
   Observation

   The left matrix is [1 1 -1 | -2; 0 -3 3 | 9; 0 0 4 | 8] and the right matrix becomes [1 1 -1 | 2; 0 1 -1 | -3; 0 0 1 | 2].

3. Audio
   Observation

   The speaker verbally computes each scaled entry.

1. The first-row constant is initially written as 2 on the right-hand matrix, but the speaker later corrects it to -2.

1. Expression

   $$\begin{bmatrix}1&1&-1&|&-2\\0&-3&3&|&9\\0&0&4&|&8\end{bmatrix}$$
   Explanation

   Start from the given upper-triangular augmented matrix.

   Justification

   Observed directly on screen.

   Shown in the video
2. Expression

   $$R_2 \leftarrow -\frac{1}{3}R_2$$
   Explanation

   Scale the entire second row by -1/3 to make its pivot equal to 1.

   Justification

   Stated by the speaker and written above the arrow.

   Shown in the video
3. Expression

   $$\begin{bmatrix}0&1&-1&|&-3\end{bmatrix}$$
   Explanation

   The second row becomes 0, 1, -1 | -3 because -3(-1/3)=1, $3(-1/3)=-1$, and $9(-1/3)=-3$.

   Justification

   Arithmetic explicitly spoken and shown.

   Shown in the video
4. Expression

   $$R_3 \leftarrow \frac{1}{4}R_3$$
   Explanation

   Scale the entire third row by $1/4$ to make its pivot equal to 1.

   Justification

   Stated by the speaker and written below the arrow.

   Shown in the video
5. Expression

   $$\begin{bmatrix}0&0&1&|&2\end{bmatrix}$$
   Explanation

   The third row becomes 0, 0, 1 | 2 because $4(1/4)=1$ and $8(1/4)=2$.

   Justification

   Arithmetic explicitly spoken and shown.

   Shown in the video
6. Expression

   $$\begin{bmatrix}1&1&-1&|&-2\\0&1&-1&|&-3\\0&0&1&|&2\end{bmatrix}$$
   Explanation

   After correcting the sign transfer in the first row, the final augmented matrix is in the displayed row echelon form.

   Justification

   Combines the visible result with the later spoken correction that the first-row constant should be -2.

   Derived from the video

The original augmented matrix is converted to a unit-pivot upper-triangular augmented matrix suitable for back substitution.

1. Diagram
   Observation

   The equations $x + y - z = -2$, $y - z = -3$, $z = 2$ are written, then solved step by step.

2. Audio
   Observation

   The speaker narrates substituting z into the second equation and then y and z into the first.

1. Expression

   $$z = 2$$
   Explanation

   The third equation directly gives the value of z.

   Justification

   Read from the last row of the echelon-form matrix.

   Shown in the video
2. Expression

   $$y - z = -3$$
   Explanation

   Use the second equation before substitution.

   Justification

   Read from the second row of the echelon-form matrix.

   Shown in the video
3. Expression

   $$y - 2 = -3$$
   Explanation

   Substitute $z = 2$ into the second equation.

   Justification

   Explicitly stated by the speaker.

   Shown in the video
4. Expression

   $$y = -1$$
   Explanation

   Add 2 to both sides: -3 + $2 = -1$.

   Justification

   Algebraic simplification shown and spoken.

   Shown in the video
5. Expression

   $$x + y - z = -2$$
   Explanation

   Use the first equation before substitution.

   Justification

   Read from the first row of the echelon-form matrix after sign correction.

   Shown in the video
6. Expression

   $$x + (-1) - (2) = -2$$
   Explanation

   Substitute $y = -1$ and $z = 2$ into the first equation.

   Justification

   Explicitly written on screen.

   Shown in the video
7. Expression

   $$x - 3 = -2$$
   Explanation

   Combine constants: -1 - $2 = -3$.

   Justification

   Spoken arithmetic and visible simplification.

   Shown in the video
8. Expression

   $$x = 1$$
   Explanation

   Add 3 to both sides: -2 + $3 = 1$.

   Justification

   Final algebraic step shown and boxed.

   Shown in the video

The solution of the system is $x = 1$, $y = -1$, $z = 2$.

1. Formula
   Observation

   Visible equations include $y - z = -3$, $z = 2$, and $x + (-1) - (2) = -2$.

2. Diagram
   Observation

   Colored boxes show $y = -1$, $z = 2$, $x = 1$, and the final triple (1,-1,2).

3. Audio
   Observation

   'Now we have the final answer... in the form of x, y, z, it's going to be 1, negative 1, comma 2.'

1. Expression

   $$z = 2$$
   Explanation

   The bottom equation gives z directly.

   Justification

   Read from the displayed equation $z = 2$.

   Shown in the video
2. Expression

   $$y - z = -3$$
   Explanation

   Substitute $z = 2$ into the second equation to solve for y.

   Justification

   Displayed equation $y - z = -3$ together with $z = 2$.

   Shown in the video
3. Expression

   $$y = -1$$
   Explanation

   Solving $y - 2 = -3$ yields $y = -1$.

   Justification

   Algebraic simplification of the previous step.

   Derived from the video
4. Expression

   $$x + y - z = -2$$
   Explanation

   Substitute $y = -1$ and $z = 2$ into the first equation to solve for x.

   Justification

   Displayed equation $x + y - z = -2$ and previously found values.

   Shown in the video
5. Expression

   $$x + (-1) - (2) = -2$$
   Explanation

   The substitution is written explicitly on screen.

   Justification

   Directly visible in the worked solution.

   Shown in the video
6. Expression

   $$x = 1$$
   Explanation

   Solving $x - 3 = -2$ gives $x = 1$.

   Justification

   Algebraic simplification shown in the board work.

   Derived from the video
7. Expression

   $$(1, -1, 2)$$
   Explanation

   The ordered triple records the solution in the order x, y, z.

   Justification

   Speaker states the final answer in this form and the board shows the boxed values.

   Shown in the video

The first system has solution (x,y,z)=(1,-1,2).

1. Formula
   Observation

   Initial augmented matrix is [[2,1,-1|1],[3,2,1|10],[2,-1,2|6]].

2. Formula
   Observation

   Operation label $R_3 - R_1$ is written next to the arrow.

3. Audio
   Observation

   Speaker computes $2-2=0$, $1-(-1)=2$, -1-2=-3, and $1-6=-5$.

4. Diagram
   Observation

   Resulting third row is written as [0, 2, -3 | -5].

1. Expression

   $$\begin{bmatrix}2&1&-1&|&1\\3&2&1&|&10\\2&-1&2&|&6\end{bmatrix}$$
   Explanation

   Start from the augmented matrix of the new system.

   Justification

   Matrix is written directly from the equations.

   Shown in the video
2. Expression

   $$R_3 \leftarrow R_3 - R_1$$
   Explanation

   Replace row 3 by row 3 minus row 1 to eliminate the first entry in row 3.

   Justification

   Explicit operation label and spoken explanation.

   Shown in the video
3. Expression

   $$2-2=0$$
   Explanation

   First entry of the new row 3.

   Justification

   Column-wise subtraction shown and spoken.

   Shown in the video
4. Expression

   $$1-(-1)=2$$
   Explanation

   Second entry of the new row 3.

   Justification

   Column-wise subtraction shown and spoken.

   Shown in the video
5. Expression

   $$-1-2=-3$$
   Explanation

   Third entry of the new row 3.

   Justification

   Column-wise subtraction shown and spoken.

   Shown in the video
6. Expression

   $$1-6=-5$$
   Explanation

   Augmented constant entry of the new row 3.

   Justification

   Column-wise subtraction shown and spoken.

   Shown in the video
7. Expression

   $$\begin{bmatrix}2&1&-1&|&1\\3&2&1&|&10\\0&2&-3&|&-5\end{bmatrix}$$
   Explanation

   Rows 1 and 2 remain unchanged while row 3 becomes [0,2,-3|-5].

   Justification

   Speaker says everything else stays the same and the board shows the updated matrix.

   Shown in the video