Linear System

Multiple linear equations, one shared goal: finding consistent solutions across variables. Linear systems reveal structure in problems and form the basis for scalable algorithms in science and engineering. 🔄

Systems of Linear Equations

A system of linear equations is a collection of one or more linear equations involving the same set of variables \(x_1, x_2, \ldots, x_n\).

\[ \begin{aligned} a_1 x_1 + b_1 x_2 &= c_1 \\ a_2 x_1 + b_2 x_2 &= c_2 \\ \end{aligned} \]

viewof a1 = Inputs.range([-10, 10], {
  step: 1, 
  value: 2, 
  label: tex`a_1`, 
  width: 200
})

viewof b1 = Inputs.range([-10, 10], {
  step: 1, 
  value: 3, 
  label: tex`b_1`, 
  width: 200
})

viewof c1 = Inputs.range([-20, 20], {
  step: 1, 
  value: 6, 
  label: tex`c_1`, 
  width: 200
})

// Second equation parameters
viewof a2 = Inputs.range([-10, 10], {
  step: 1, 
  value: 1, 
  label: tex`a_2`, 
  width: 200
})

viewof b2 = Inputs.range([-10, 10], {
  step: 1, 
  value: -2, 
  label: tex`b_2`, 
  width: 200
})

viewof c2 = Inputs.range([-20, 20], {
  step: 1, 
  value: 4, 
  label: tex`c_2`, 
  width: 200
})

// Display both equations
html`<div style="text-align: center; font-size: 1.2em; margin: 1em 0;">
  <p>${tex`${a1}x_1 ${b1 >= 0 ? "+" : "-"} ${Math.abs(b1)}x_2 = ${c1}`}</p>
  <p>${tex`${a2}x_1 ${b2 >= 0 ? "+" : "-"} ${Math.abs(b2)}x_2 = ${c2}`}</p>
</div>`

solution = {
  // Using Cramer's rule to solve the system
  const det = a1 * b2 - a2 * b1;
  
  if (Math.abs(det) < 1e-10) {
    // Check if lines are parallel and coincident
    const k = a1 !== 0 ? a2/a1 : b2/b1;
    if (Math.abs(c2 - k * c1) < 1e-10) {
      return { status: "Infinite solutions", x1: null, x2: null };
    } else {
      return { status: "No solution", x1: null, x2: null };
    }
  } else {
    // Unique solution
    const x1 = (c1 * b2 - c2 * b1) / det;
    const x2 = (a1 * c2 - a2 * c1) / det;
    return { status: "Unique solution", x1, x2 };
  }
}

// Generate points for first line
points1 = {
  const result = [];
  for (let i = 0; i < 200; i++) {
    const x = i / 10 - 10;
    if (b1 !== 0) {
      const y = (c1 - a1 * x) / b1;
      if (y >= -10 && y <= 10) {
        result.push({x, y});
      }
    }
  }
  return result;
}

// Generate points for second line
points2 = {
  const result = [];
  for (let i = 0; i < 200; i++) {
    const x = i / 10 - 10;
    if (b2 !== 0) {
      const y = (c2 - a2 * x) / b2;
      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: "x₁", domain: [-10, 10]},
  y: {label: "x₂", domain: [-10, 10]},
  marks: [
    Plot.ruleY([0]),
    Plot.ruleX([0]),
    Plot.line(points1, {
      x: "x", 
      y: "y", 
      stroke: "steelblue",
      strokeWidth: 2
    }),
    Plot.line(points2, {
      x: "x", 
      y: "y", 
      stroke: "crimson",
      strokeWidth: 2
    }),
    solution.x1 !== null ? Plot.dot([{x: solution.x1, y: solution.x2}], {
      x: "x",
      y: "y",
      r: 6,
      fill: "black",
      stroke: "white",
      strokeWidth: 2
    }) : null
  ]
})

html`<div style="text-align: center; margin: 1em 0; padding: 10px; border-radius: 5px; background-color: ${solution.status === "No solution" ? "#ffecec" : solution.status === "Infinite solutions" ? "#fffacd" : "#e6ffe6"};">
  <p style="font-weight: bold;"> ${solution.status}</p>
  ${solution.x1 !== null ? html`<p>${tex`(x_1, x_2) = (${solution.x1.toFixed(2)}, ${solution.x2.toFixed(2)})`}</p>` : ''}
</div>`