数学 – Python - JavaScript - 数学はここから始まる - 数 – 高次方程式 – 3元以上の連立1次方程式(未知数の消去、工夫した方法、平面、空間の直線の交点)

数学読本〈1〉
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• Nebo(Windows アプリ)
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• 参考書籍
  • Pythonからはじめる数学入門 (楽天ブックス(Kobo) Yahoo!)
  • JavaScriptリファレンス 第6版 (楽天ブックス(Kobo) Yahoo!)
  • インタラクティブ・データビジュアライゼーション (楽天ブックス(Kobo) Yahoo!)

数学読本〈1〉数・式の計算/方程式/不等式 (松坂 和夫(著)、岩波書店)の第3章(数学の威力を発揮する - 方程式)、3.3(高次方程式)、3元以上の連立1次方程式の問30.を取り組んでみる。

30. 
    
    1. 
       
       $\begin{array}{}5x=15\\ x=3\\ 2y=-4\\ y=-2\\ x=3,y=-2\end{array}$
    2. 
       
       $\begin{array}{}-11y=-44\\ y=4\\ x=27-24=3\end{array}$
    3. 
       
       $\begin{array}{}-4x-11y=18.\\ 9x+12y-6z=-33\\ 8x-4y+6z=34\\ 17x+8y=1\\ -68x-187y=306\\ 68x+32y=4\\ -155y=310\\ y=-2\\ x=1\\ 3-8-2z=-11\\ z=3\end{array}$
    4. 
       
       $\begin{array}{}5x-2y=-25\\ 2x+5y=19\\ 10x-4y=-50\\ 10x+25y=95\\ 29y=145\\ y=5\\ 2x+25=19\\ x=-3\\ -6+5-z=0\\ z=-1\\ x=-3,y=5,z=-1\end{array}$
    5. 
       
       $\begin{array}{}3x+2y=8\\ 3y+z=10\\ 5x+y-z=12\\ z=10-3y\\ 5x+y-10+3y=12\\ 5x+4y=22\\ -x=6\\ x=-6\\ -18+2y=8\\ y=13\\ z=10-39=-29\end{array}$
    6. 
       
       $\begin{array}{}3\left(x+y+z+u\right)=30\\ x+y+z+u=10\\ x=10-9=1\\ y=10-8=2\\ z=10-7=3\\ u=10-6=4\end{array}$
    7. 
       
       $\begin{array}{}6\left(x+y+z+u\right)=0\\ x+y+z+u=0\\ 2x=5\\ x=\frac{5}{2}\\ y=0\\ z=-4\\ u=\frac{3}{2}\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, solve, sqrt, I, plot

print('30.')

x, y, z, u = symbols('x, y, z, u')
eqss = [(x + 2 * y + 1, 3 * x - 4 * y - 17),
        (2 * x + y - 10, x + 6 * y - 27),
        (2 * x - 3 * y - 4 * z + 4,
         3 * x + 4 * y - 2 * z + 11,
         4 * x - 2 * y + 3 * z - 17),
        (x - 4 * y + 2 * z + 25,
         2 * x + y - z,
         3 * x + y + 2 * z + 6),
        ((3 * x + 2 * y) / 4 - 2,
         (3 * y + z) / 5 - 2,
         (5 * x + y - z) / 6 - 2),
        (y + z + u - 9,
         z + u + x - 8,
         u + x + y - 7,
         x + y + z - 6),
        (x + y + z + 3 * u - 3,
         x + y + 3 * z + u + 8,
         x + 3 * y + z + u,
         3 * x + y + z + u - 5)]

for i, eqs in enumerate(eqss, 1):
    print(f'({i})')
    pprint(solve(eqs, dict=True))
    print()

p = plot((-1 - x) / 2, (17 - 3 * x) / (-4),
         ylim=(-10, 10), show=False, legend=True)
colors = ['red', 'green']

for i, color in enumerate(colors):
    p[i].line_color = color

p.save('sample30.png')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample30.py
30.
(1)
[{x: 3, y: -2}]

(2)
[{x: 3, y: 4}]

(3)
[{x: 1, y: -2, z: 3}]

(4)
[{x: -3, y: 5, z: -1}]

(5)
[{x: -6, y: 13, z: -29}]

(6)
[{u: 4, x: 1, y: 2, z: 3}]

(7)
[{u: 3/2, x: 5/2, y: 0, z: -4}]

$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="1">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.001" value="0.005">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-10">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-10">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">

<button id="draw0">draw</button>
<button id="clear0">clear</button>

<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script>

<script src="sample30.js"></script>

JavaScript

let div0 = document.querySelector('#graph0'),
    pre0 = document.querySelector('#output0'),
    width = 600,
    height = 600,
    padding = 50,
    btn0 = document.querySelector('#draw0'),
    btn1 = document.querySelector('#clear0'),
    input_r = document.querySelector('#r0'),
    input_dx = document.querySelector('#dx'),
    input_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    input_y1 = document.querySelector('#y1'),
    input_y2 = document.querySelector('#y2'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    p = (x) => pre0.textContent += x + '\n';

let fns = [[x => (-1 - x) / 2, 'red'],
           [x => (17 - 3  * x) / (-4), 'green']];
    
let draw = () => {
    pre0.textContent = '';

    let r = parseFloat(input_r.value),
        dx = parseFloat(input_dx.value),
        x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }    

    let points = [],
        lines = [[3, y1, 3, y2, 'blue'],
                 [x1, -2, x2, -2, 'brown']];
    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                points.push([x, y, color]);
            }
        });
    
    let xscale = d3.scaleLinear()
        .domain([x1, x2])
        .range([padding, width - padding]);
    let yscale = d3.scaleLinear()
        .domain([y1, y2])
        .range([height - padding, padding]);

    let xaxis = d3.axisBottom().scale(xscale);
    let yaxis = d3.axisLeft().scale(yscale);
    div0.innerHTML = '';
    let svg = d3.select('#graph0')
        .append('svg')
        .attr('width', width)
        .attr('height', height);

    svg.selectAll('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
        .enter()
        .append('line')
        .attr('x1', (d) => xscale(d[0]))
        .attr('y1', (d) => yscale(d[1]))
        .attr('x2', (d) => xscale(d[2]))
        .attr('y2', (d) => yscale(d[3]))
        .attr('stroke', (d) => d[4] || 'black');

    svg.selectAll('circle')
        .data(points)
        .enter()
        .append('circle')
        .attr('cx', (d) => xscale(d[0]))
        .attr('cy', (d) => yscale(d[1]))
        .attr('r', r)
        .attr('fill', (d) => d[2] || 'green');
    
    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);

    [fns].forEach((fs) => p(fs.join('\n')));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();

r = dx =
x1 = x2 =
y1 = y2 =