Algebra / English

Gaussian Elimination

Leios · YouTube · 5:45

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IN THIS VIDEO

01Use an augmented matrix
02Preserve the solution set
03Back-substitute and verify

SELECTED MOMENT

Use an augmented matrix

SUGGESTED PREPARATION

Linear equations and augmented matrices

CONCEPT CONNECTIONS

Gaussian eliminationExplanation
Why this connection? · 0:58
The system $2x+3y+4z=6$ , $x+2y+3z=4$ , $3x-4y=10$ is encoded by its coefficients and a constants column. An echelon form creates a staircase of pivots. The permitted elementary operations are row swapping, multiplication of a row by a nonzero scalar, and adding a multiple of one row to another. Each produces an equivalent system. The triangular system contains $11z=18$ and $y+2z=2$ . Back substitution gives $z=18/11$ , $y=-14/11$ , $x=18/11$ . Substitution into the three original equations confirms the result. Each equation in three variables represents a plane. In this example, elimination changes the planes while preserving their common intersection point. Continuing to reduced row echelon form makes the solution readable directly.
Gaussian eliminationExplanation
Why this connection? · 0:58
The system $2x+3y+4z=6$ , $x+2y+3z=4$ , $3x-4y=10$ is encoded by its coefficients and a constants column. An echelon form creates a staircase of pivots. The permitted elementary operations are row swapping, multiplication of a row by a nonzero scalar, and adding a multiple of one row to another. Each produces an equivalent system. The triangular system contains $11z=18$ and $y+2z=2$ . Back substitution gives $z=18/11$ , $y=-14/11$ , $x=18/11$ . Substitution into the three original equations confirms the result. Each equation in three variables represents a plane. In this example, elimination changes the planes while preserving their common intersection point. Continuing to reduced row echelon form makes the solution readable directly.

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Read the full overview

Gaussian elimination replaces a linear system with an equivalent, easier system. The lesson connects row operations, back substitution and moving planes whose common intersection is preserved.

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Chapters

0:58Use an augmented matrix 1:30Preserve the solution set 2:08Back-substitute and verify 3:11The geometric meaning

Learning script

Generated from the video's visuals and explanation; not verbatim speech.

The system $2x+3y+4z=6$ , $x+2y+3z=4$ , $3x-4y=10$ is encoded by its coefficients and a constants column. An echelon form creates a staircase of pivots.

The permitted elementary operations are row swapping, multiplication of a row by a nonzero scalar, and adding a multiple of one row to another. Each produces an equivalent system.

The triangular system contains $11z=18$ and $y+2z=2$ . Back substitution gives $z=18/11$ , $y=-14/11$ , $x=18/11$ . Substitution into the three original equations confirms the result.

Each equation in three variables represents a plane. In this example, elimination changes the planes while preserving their common intersection point. Continuing to reduced row echelon form makes the solution readable directly.

Knowledge cards

Use an augmented matrix

The system $2x+3y+4z=6$ , $x+2y+3z=4$ , $3x-4y=10$ is encoded by its coefficients and a constants column. An echelon form creates a staircase of pivots.

Preserve the solution set

The permitted elementary operations are row swapping, multiplication of a row by a nonzero scalar, and adding a multiple of one row to another. Each produces an equivalent system.

Back-substitute and verify

The triangular system contains $11z=18$ and $y+2z=2$ . Back substitution gives $z=18/11$ , $y=-14/11$ , $x=18/11$ . Substitution into the three original equations confirms the result.

The geometric meaning

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Gaussian elimination ExplanationAt 0:58
Why this connection?
The system $2x+3y+4z=6$ , $x+2y+3z=4$ , $3x-4y=10$ is encoded by its coefficients and a constants column. An echelon form creates a staircase of pivots. The permitted elementary operations are row swapping, multiplication of a row by a nonzero scalar, and adding a multiple of one row to another. Each produces an equivalent system. The triangular system contains $11z=18$ and $y+2z=2$ . Back substitution gives $z=18/11$ , $y=-14/11$ , $x=18/11$ . Substitution into the three original equations confirms the result. Each equation in three variables represents a plane. In this example, elimination changes the planes while preserving their common intersection point. Continuing to reduced row echelon form makes the solution readable directly.
Gaussian elimination ExplanationAt 0:58
Why this connection?
The system $2x+3y+4z=6$ , $x+2y+3z=4$ , $3x-4y=10$ is encoded by its coefficients and a constants column. An echelon form creates a staircase of pivots. The permitted elementary operations are row swapping, multiplication of a row by a nonzero scalar, and adding a multiple of one row to another. Each produces an equivalent system. The triangular system contains $11z=18$ and $y+2z=2$ . Back substitution gives $z=18/11$ , $y=-14/11$ , $x=18/11$ . Substitution into the three original equations confirms the result. Each equation in three variables represents a plane. In this example, elimination changes the planes while preserving their common intersection point. Continuing to reduced row echelon form makes the solution readable directly.

Gaussian Elimination

The explanation, unpacked.

Chapters

Learning script

Knowledge cards

Use an augmented matrix

Preserve the solution set

Back-substitute and verify

The geometric meaning

Explore the knowledge in this video

Similar explanations

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