数学 - Python - 代数学 - 連立方程式と高次方程式 - 連立2次方程式 - 連立2元2次方程式、未知数、解

学習環境

代数への出発 (新装版数学入門シリーズ) (松坂和夫(著)、岩波書店)の第5章(連立方程式と高次方程式)、2(連立2次方程式)、問9の解答を求めてみる。

1. $\begin{array}{l} x^{2} + {(x + 1)}^{2} = 13 \\ 2 x^{2} + 2 x - 12 = 0 \\ x^{2} + x - 6 = 0 \\ (x + 3) (x - 2) = 0 \\ x = - 3, 2 \end{array}$
  
  よって解は、
  
  $\begin{array}{l} {\begin{cases} x = - 3 \\ y = - 2 \end{cases} \\ {\begin{cases} x = 2 \\ y = 3 \end{cases} \end{array}$
2. $\begin{array}{l} y = - x - 5 \\ x (- x - 5) = 6 \\ x^{2} + 5 x + 6 = 0 \\ (x + 2) (x + 3) = 0 \\ {\begin{cases} x = - 2 \\ y = - 3 \end{cases} \\ {\begin{cases} x = - 3 \\ y = - 2 \end{cases} \end{array}$
3. $\begin{array}{l} y = 5 - x \\ x^{2} + x (5 - x) + {(5 - x)}^{2} = 21 \\ x^{2} - 5 x + 4 = 0 \\ (x - 1) (x - 4) = 0 \\ {\begin{cases} x = 1 \\ y = 4 \end{cases} \\ {\begin{cases} x = 4 \\ y = 1 \end{cases} \end{array}$
4. $\begin{array}{l} y = x - 8 \\ x (x - 8) = - 15 \\ x^{2} - 8 x + 15 = 0 \\ (x - 3) (x - 5) = 0 \\ {\begin{cases} x = 3 \\ y = - 5 \end{cases} \\ {\begin{cases} x = 5 \\ y = - 3 \end{cases} \end{array}$
5. $\begin{array}{l} {(2 x - 1)}^{2} - x^{2} = 5 \\ 3 x^{2} - 4 x - 4 = 0 \\ (x - 2) (3 x + 2) = 0 \\ {\begin{cases} x = - \frac{2}{3} \\ y = - \frac{7}{3} \end{cases} \\ {\begin{cases} x = 2 \\ y = 3 \end{cases} \end{array}$
6. $\begin{array}{l} y = \frac{5 - 3 x}{4} \\ 4 x^{2} + x \cdot \frac{5 - 3 x}{4} - 3 \cdot {(\frac{5 - 3 x}{4})}^{2} = 0 \\ 64 x^{2} + 20 x - 12 x^{2} - 3 (25 - 30 x + 9 x^{2}) = 0 \\ 25 x^{2} + 110 x - 75 = 0 \\ 5 x^{2} + 22 x - 15 = 0 \\ (x + 5) (5 x - 3) = 0 \\ {\begin{cases} x = - 5 \\ y = 5 \end{cases} \\ {\begin{cases} x = \frac{3}{5} \\ y = \frac{4}{5} \end{cases} \end{array}$
7. $\begin{array}{l} (x - 2 y) (x - 3 y) = 0 \\ x = 2 y \\ 4 y^{2} + 2 y^{2} + 2 y^{2} = 56 \\ y^{2} = 7 \\ y = \pm \sqrt{7} \\ x = 3 y \\ 9 y^{2} + 3 y^{2} + 2 y^{2} = 56 \\ y^{2} = 4 \\ y = \pm 2 \\ {\begin{cases} x = - 2 \sqrt{7} \\ y = - \sqrt{7} \end{cases} \\ {\begin{cases} x = 2 \sqrt{7} \\ y = \sqrt{7} \end{cases} \\ {\begin{cases} x = - 6 \\ y = - 2 \end{cases} \\ {\begin{cases} x = 6 \\ y = 2 \end{cases} \end{array}$
8. $\begin{array}{l} (x - 2 y) (x - 18 y) = 0 \\ x = 2 y \\ 4 y^{2} - 4 y^{2} + 6 y^{2} = 6 \\ y = \pm 1 \\ x = 18 y \\ 1 8^{2} y^{2} - 2 \cdot 18 \cdot y^{2} + 6 y^{2} = 6 \\ 3 \cdot 18 y^{2} - 2 \cdot 3 y^{2} + y^{2} = 1 \\ y^{2} = \frac{1}{49} \\ y = \pm \frac{1}{7} \\ {\begin{cases} x = - 2 \\ y = - 1 \end{cases} \\ {\begin{cases} x = 2 \\ y = 1 \end{cases} \\ {\begin{cases} x = - \frac{18}{7} \\ y = - \frac{1}{7} \end{cases} \\ {\begin{cases} x = \frac{18}{7} \\ y = \frac{1}{7} \end{cases} \end{array}$
9. $\begin{array}{l} 9 x y + 27 y^{2} = 12 \\ 9 x y + 27 y^{2} = x^{2} + 3 x y \\ x^{2} - 6 x y - 27 y^{2} = 0 \\ (x - 9 y) (x + 3 y) = 0 \\ x = 9 y \\ 81 y^{2} + 27 y^{2} = 12 \\ y^{2} = \frac{12}{108} = \frac{1}{9} \\ y = \pm \frac{1}{3} \\ x = - 3 y \\ 9 y^{2} - 9 y^{2} \neq 12 \\ {\begin{cases} x = - 3 \\ y = - \frac{1}{3} \end{cases} \\ {\begin{cases} x = 3 \\ y = \frac{1}{3} \end{cases} \end{array}$
10. $\begin{array}{l} {\begin{cases} 60 x y + 48 x - 72 y - 60 = 0 \\ 60 x y + 45 x - 70 y - 55 = 0 \end{cases} \\ 3 x - 2 y - 5 = 0 \\ x = \frac{2 y + 5}{3} \\ 8 y^{2} + 20 y + 6 y + 15 - 14 y - 11 = 0 \\ 8 y^{2} + 12 y + 4 = 0 \\ 2 y^{2} + 3 y + 1 = 0 \\ (y + 1) (2 y + 1) = 0 \\ {\begin{cases} x = 1 \\ y = - 1 \end{cases} \\ {\begin{cases} x = \frac{4}{3} \\ y = - \frac{1}{2} \end{cases} \end{array}$
11. $\begin{array}{l} 2 x y - x - y = x y - 1 \\ x y - x - y + 1 = 0 \\ (x - 1) y - (x - 1) = 0 \\ (x - 1) (y - 1) = 0 \\ x = 1 \\ 1 + y^{2} = 5 (y - 1) \\ y^{2} - 5 y + 6 = 0 \\ (y - 2) (y - 3) = 0 \\ y = 2, 3 \\ y = 1 \\ x = 2, 3 \\ {\begin{cases} x = 1 \\ y = 2 \end{cases} \\ {\begin{cases} x = 1 \\ y = 3 \end{cases} \\ {\begin{cases} x = 2 \\ y = 1 \end{cases} \\ {\begin{cases} x = 3 \\ y = 1 \end{cases} \end{array}$

#!/usr/bin/env python3 from unittest import TestCase, main from sympy import symbols, solve, Rational, sqrt print('9.') x, y = symbols('x, y') class TestEquations(TestCase): def test1(self): self.assertEqual(solve([x ** 2 + y ** 2 - 13, y - x - 1]), [{x: -3, y: -2}, {x: 2, y: 3}]) def test2(self): self.assertEqual(solve([x + y + 5, x * y - 6]), [{x: -3, y: -2}, {x: -2, y: -3}]) def test3(self): self.assertEqual(solve([x + y - 5, x ** 2 + x * y + y ** 2 - 21]), [{x: 1, y: 4}, {x: 4, y: 1}]) def test4(self): self.assertEqual(solve([x - y - 8, x * y + 15]), [{x: 3, y: -5}, {x: 5, y: -3}]) def test5(self): self.assertEqual(solve([y - 2 * x + 1, y ** 2 - x ** 2 - 5]), [{x: -Rational(2, 3), y: -Rational(7, 3)}, {x: 2, y: 3}]) def test6(self): self.assertEqual(solve([3 * x + 4 * y - 5, 4 * x ** 2 + x * y - 3 * y ** 2]), [{x: -5, y: 5}, {x: Rational(3, 5), y: Rational(4, 5)}]) def test7(self): self.assertEqual( sorted(solve([x ** 2 + x * y + 2 * y ** 2 - 56, x ** 2 - 5 * x * y + 6 * y ** 2]), key=lambda d: d[x]), sorted([{x: -2 * sqrt(7), y: -sqrt(7)}, {x: 2 * sqrt(7), y: sqrt(7)}, {x: -6, y: -2}, {x: 6, y: 2}], key=lambda d: d[x])) def test8(self): self.assertEqual( sorted(solve([x ** 2 - 20 * x * y + 36 * y ** 2, x ** 2 - 2 * x * y + 6 * y ** 2 - 6]), key=lambda d: d[x]), sorted([{x: -2, y: -1}, {x: 2, y: 1}, {x: -Rational(18, 7), y: -Rational(1, 7)}, {x: Rational(18, 7), y: Rational(1, 7)}], key=lambda d: d[x])) def test9(self): self.assertEqual( sorted(solve([x ** 2 + 3 * x * y - 12, x * y + 3 * y ** 2 - Rational(4, 3)]), key=lambda d: d[x]), sorted([{x: -3, y: -Rational(1, 3)}, {x: 3, y: Rational(1, 3)}], key=lambda d: d[x]) ) def test10(self): self.assertEqual( sorted(solve([5 * x * y + 4 * x - 6 * y - 5, 12 * x * y + 9 * x - 14 * y - 11]), key=lambda d: d[x]), sorted([{x: 1, y: -1}, {x: Rational(4, 3), y: -Rational(1, 2)}], key=lambda d: d[x]) ) def test11(self): self.assertEqual( sorted(solve([x ** 2 + y ** 2 - 10 * x * y + 5 * (x + y), x ** 2 + y ** 2 - 5 * (x * y - 1)]), key=lambda d: d[x]), sorted([{x: 1, y: 2}, {x: 1, y: 3}, {x: 2, y: 1}, {x: 3, y: 1}], key=lambda d: d[x]) ) if __name__ == "__main__": main()

% ./sample9.py -v 9. test1 (__main__.TestEquations) ... ok test10 (__main__.TestEquations) ... ok test11 (__main__.TestEquations) ... ok test2 (__main__.TestEquations) ... ok test3 (__main__.TestEquations) ... ok test4 (__main__.TestEquations) ... ok test5 (__main__.TestEquations) ... ok test6 (__main__.TestEquations) ... ok test7 (__main__.TestEquations) ... ok test8 (__main__.TestEquations) ... ok test9 (__main__.TestEquations) ... ok ---------------------------------------------------------------------- Ran 11 tests in 0.519s OK %

Kamimura's blog

2020年4月12日日曜日

数学 - Python - 代数学 - 連立方程式と高次方程式 - 連立2次方程式 - 連立2元2次方程式、未知数、解

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