Algebra and Conics – Linear Systems, Induction & Binomial Theorem, Conic Sections
MathematicsAlgebraAges 15–16
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Sign in to playThree math topics in one simulation, chosen with the topic parameter. System of three linear equations: step through Gaussian elimination on the coefficient matrix to see whether the system has a unique solution, no solution, or infinitely many solutions. Induction and the binomial theorem: click cells of Pascal's triangle to see the addition rule and the expansion of (a+b)^n, with a staircase diagram proving 1+2+…+n = n(n+1)/2. Conic sections: drag point M along the curve, check that MF = e·d(M, Δ), and watch the ellipse, parabola, and hyperbola change as the eccentricity e varies.
Lesson: Systems of three linear equations; mathematical induction and the binomial theorem; the three conic sections and their applications
What it shows
Three topics share one page. A system of three linear equations describes three planes; Gaussian elimination reduces it to triangular form, and the last row shows whether the planes meet in one point, have no common point (a row reading 0 = 1) or share infinitely many points (a row reading 0 = 0). Pascal's triangle follows the rule C(n; k) = C(n − 1; k − 1) + C(n − 1; k), gives the coefficients of (a + b)ⁿ and supports proof by induction. The conics are defined together as the points M with MF = e·d(M, Δ): an ellipse if e < 1, a parabola if e = 1, a hyperbola if e > 1.
How to use
Choose the Current topic. For the system, pick an Example system of three equations, click Next step for each elimination and read the last row; Switch system compares the three cases. For Pascal's triangle, click Add row (n + 1) and click a cell to see the two numbers it comes from. For conics, change Eccentricity e of the conic or click Increase e, and drag point M to check that MF/d(M, Δ) stays equal to e.
Parameters you can change
- Current topic 1. System of three linear equations, 2. Mathematical induction and the binomial theorem, 3. Conic sections and applications
- Example system of three equations System with a unique solution, System with no solution, System with infinitely many solutions
- Exponent n of (a + b)ⁿ and number of terms in the sum 1–10
- Eccentricity e of the conic 0.2–2.4
- Show detailed labels and values
Questions to explore
- Which row of the reduced matrix shows that a system of three linear equations has no solution?
- What is the sum of the numbers in row n of Pascal's triangle, and why?
- For which value of e is the conic a parabola, and what happens to the curve as e passes it?
