Lines

Straightforward yet powerful—lines represent the simplest form of linear relationships. They’re the geometric starting point for understanding equations, slopes, and intersections in \(\mathbb{R}^2\) and beyond. 📈

Linear Equations

A linear equation in the variables \(x_1, x_2, \ldots, x_n\) is an equation that can be written in the form:

\[ a_1x_1 + a_2x_2 + \cdots + a_nx_n = b \]

\(b \in \mathbb{R}\) is a constant, \(a_1, a_2, \ldots, a_n \in \mathbb{R}\) are coefficients typically known in advance, and \(n\) is a positive integer indicating the number of variables.

viewof a = Inputs.range([-10, 10], {
  step: 1, 
  value: 2, 
  label: tex`a_1`, 
  width: 200
})

viewof b = Inputs.range([-10, 10], {
  step: 1, 
  value: 3, 
  label: tex`a_2`, 
  width: 200
})

viewof c = Inputs.range([-20, 20], {
  step: 1, 
  value: 6, 
  label: tex`b`, 
  width: 200
})

// Display the equation using html and tex literals
html`<div style="text-align: center; font-size: 1.2em; margin: 1em 0;">
  ${tex`${a}x_1 ${b >= 0 ? "+" : "-"} ${Math.abs(b)}x_2 = ${c}`}
</div>`

points = {
  const result = [];
  for (let i = 0; i < 200; i++) {
    const x = i / 10 - 10;
    if (b !== 0) {
      const y = (c - a * x) / b;
      if (y >= -10 && y <= 10) {
        result.push({x, y});
      }
    }
  }
  return result;
}

// Create the plot
Plot.plot({
  style: "overflow: visible; display: block; margin: 0 auto;",
  width: 500,
  height: 400,
  grid: true,
  x: {label: "x_1", domain: [-10, 10]},
  y: {label: "x_2", domain: [-10, 10]},
  marks: [
    Plot.ruleY([0]),
    Plot.ruleX([0]),
    Plot.line(points, {
      x: "x", 
      y: "y", 
      stroke: "steelblue",
      strokeWidth: 2
    })
  ]
})