This animation walks through the three elementary row operations—swapping, scaling, and adding multiples of rows—that let a linear system be simplified without changing its solutions. Using a concrete two-equation example, it demonstrates eliminating x₁ by subtracting twice the first row from the second, revealing each arithmetic step and the resulting simplified equation. It also shows the operation's reversibility, reinforcing why elimination preserves solution sets. Suited for first-year economics or algebra students building intuition for solving linear systems.
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Create a continuous middle section of an English learning video on Gaussian elimination. No introduction or farewell. Use a white background, dark text, restrained blue highlights, aligned equations and calm narration. Keep unchanged mathematics visible. Show one change at a time. Start with: “To simplify the system safely, we need operations that preserve its solutions.” Explain the three permitted row operations: 1. Swap two rows. 2. Multiply a row by a non-zero number. 3. Add a multiple of one row to another. Explain why they work: swapping changes the order; non-zero scaling can be reversed; adding a multiple of another row can also be reversed. Never present column swaps as row operations. Use these two equations: x₁ + x₂ = 3 2x₁ + 6x₂ + 4x₃ = 14 Narration: “We want to remove x₁ from the second equation. Subtracting twice the first equation does exactly that.” Show: R₂ ← R₂ − 2R₁ [2,6,4 | 14] − 2[1,1,0 | 3] = [0,4,4 | 8] Reveal the four calculations in order: 2−2=0; 6−2=4; 4−0=4; 14−6=8. Then show the resulting equation: 4x₂ + 4x₃ = 8 Narration: “The second equation now contains only two unknowns. We must apply the operation to the entire equation, including the right-hand side.” Show briefly that adding twice the first row restores the original second row. End with: “This is the central idea of elimination: create zeros to make the system easier to solve, without losing or adding solutions.” Audience: First-year economics students. Explain new notation clearly and assume no prior knowledge of Gaussian elimination. Teaching plan: Narration must use plain spoken English. Read x₁ as “x one” and R₂ as “row two”. Explain row operations in words. Keep mathematical symbols and operation notation on screen only.