This animation introduces Gaussian elimination by showing how a system of four equations in three unknowns can be rewritten as a coefficient matrix. It walks through identifying knowns and unknowns, organizing coefficients in order, and framing the guiding question of simplifying systems without changing their solutions. Useful for algebra or linear algebra students learning to translate equations into matrix form before applying row operations.
Narrated · 16:9 · Preview before teaching · automatic layout checks do not establish subject accuracy
PART 1 — INTRODUCTION AND MATRIX REPRESENTATION Read only NARRATION aloud. VISUAL instructions describe what to show. Keep variable order x₁, x₂, x₃ throughout. SCENE 1 — INTRODUCTION VISUAL: Title: “Solving Linear Systems with Gaussian Elimination” Guiding question: “How can we simplify a linear system without changing its solutions?” Brief roadmap: Equations → Matrix → Row operations → Solutions NARRATION: “Linear systems arise when several conditions must hold at the same time. In economics, these conditions might describe quantities, payments or investment positions. How can we simplify such a system without changing its solutions? In this video, you will learn to represent a system as a matrix, simplify it using Gaussian elimination and recognise whether it has one solution, no solution or infinitely many solutions.” SCENE 2 — THE EXAMPLE VISUAL: Show four equations, one per line: 8x₂ + 3x₃ = 11 2x₁ + 6x₂ + 4x₃ = 14 2x₁ + 2x₂ = 6 4x₁ + 8x₂ + 14x₃ = 30 Highlight the unknowns. Label: “4 equations, 3 unknowns”. NARRATION: “We will use this example throughout the explanation. The numbers are known. The values of x one, x two and x three are unknown. We seek values that satisfy all four equations simultaneously. Four equations and three unknowns do not automatically guarantee a unique solution. Equations can repeat information or contradict each other. We will learn how to recognise these cases.” SCENE 3 — COEFFICIENTS AND MATRIX REPRESENTATION VISUAL: Highlight the second equation. Label 2, 6 and 4 as coefficients; label 14 as the right-hand-side value. Rewrite the first equation: 0x₁ + 8x₂ + 3x₃ = 11 Build A: [0 8 3] [2 6 4] [2 2 0] [4 8 14] Show x as the column vector (x₁, x₂, x₃). Show b as the column vector (11, 14, 6, 30). Then show Ax = b. NARRATION: “The numbers multiplying the unknowns are called coefficients. In the second equation, they are two, six and four. Fourteen is the right-hand-side value. We always keep the variables in the same order. A missing variable has coefficient zero. The coefficient matrix A collects the coefficients. Each row represents an equation, and each column corresponds to an unknown. The unknowns form the vector x. The right-hand-side values form the vector b. We can therefore write the system as A x equals b.” SCENE 4 — THE AUGMENTED MATRIX VISUAL: Animate the right-hand-side vector joining A. Show the augmented matrix with a vertical separator: [0 8 3 | 11] [2 6 4 | 14] [2 2 0 | 6] [4 8 14 | 30] Briefly highlight each equation and its corresponding row. Hold the completed matrix on screen. NARRATION: “For Gaussian elimination, we place the right-hand-side values next to the coefficient matrix. This gives us the augmented matrix. It contains the same information as the original equations. We have changed the representation, not the problem or its solutions. Next, we will see which row operations simplify this matrix and why they preserve the solution set.” Audience: First-year economics students. Explain new notation clearly and assume no prior knowledge of Gaussian elimination. Teaching plan: Create Part 1 of a continuous university mathematics video about Gaussian elimination: introduction and matrix representation. Use calm English narration, a plain white background, dark text and blue highlighting. Show readable equations progressively, synchronised with the narration. No avatars, decorative pictures or background music. Follow the supplied narration and scene plan. Do not add a closing summary or farewell: the next part continues with elementary row operations.