Sum–Difference Method (System of Linear Equations) — Full Explanation + Interactive Python

1. Overview of the Sum–Difference Problem

A sum–difference problem gives you:

• The sum of two unknown numbers
• The difference between the same two numbers

Your task is to find the actual numeric values.

This appears frequently in:

• Elementary arithmetic (“Wasasan”)
• Middle school algebra
• Linear equation solving
• Word problems involving comparison

---

2. Mathematical Formulation

Let the two unknown quantities be:

• (x): Ichiro’s allowance
• (y): Jiro’s allowance

The problem states:

(1) Sum

[
x + y = S
]

(2) Difference

[
x - y = D
]

These two linear equations form a system of equations:

[
\begin{cases}
x + y = S
x - y = D
\end{cases}
]

Where:

• (S): known total amount
• (D): known difference

---

3. Why the Method Works (Derivation)

Add the two equations:

[
(x + y) + (x - y) = S + D
]

Left side simplifies:

[
2x = S + D
]

Therefore:

Solution for x

[
x = \frac{S + D}{2}
]

Subtract the second equation from the first:

[
(x + y) - (x - y) = S - D
]

Left side simplifies:

[
2y = S - D
]

So:

Solution for y

[
y = \frac{S - D}{2}
]

---

4. Why the Line-Segment Diagram (Bar Model) Works

The visualization uses two horizontal bars:

• The longer bar: (x)
• The shorter bar: (y)
• The small red segment between them: the difference (D)

|---------------- x ----------------|
|------------ y ------------|
                |--- D ---|

This visually shows:

[
x = y + D
]

When both bars are “cut in half” conceptually, you obtain:

• The overlapping length = value of (y)
• The extra segment = difference (D)

This representation helps beginners intuitively understand the algebraic structure.

---

5. What the Interactive Program Does

This Python program:

• Lets the user control
  Sum (S) and Difference (D) using sliders

• Draws bars representing (x) and (y)

• Plots the red “difference segment” between them

• Computes the exact solution using:
  [
  x = \frac{S + D}{2},\quad y = \frac{S - D}{2}
  ]

• Displays the full system of equations (typeset with math)

• Includes:

  • Reset button
  • PNG saving
  • Mathematically accurate diagram

You can interactively change numbers and watch the bars adjust—this helps build intuition.

---

6. Full Interactive Python Code (English Version)

Copy & run in Google Colab.

# ============================================================
# Program Name: sum_difference_interactive.py
# Creation Date: 202501
# Purpose: Interactive Sum–Difference (System of Equations) Visualization
# ============================================================

!pip install numpy matplotlib ipywidgets --quiet

import numpy as np
import matplotlib.pyplot as plt
import ipywidgets as widgets
from ipywidgets import HBox, VBox, Button, Output
import datetime

# ------------------------------------------------------------
# PARAM_INIT
# ------------------------------------------------------------
PARAM_INIT = {
    "SUM": 2400,   # Sum (S)
    "DIFF": 400    # Difference (D)
}

# ------------------------------------------------------------
# Compute solution for x and y
# ------------------------------------------------------------
def compute_values(S, D):
    """
    Inputs:
        S : sum
        D : difference
    Outputs:
        x : larger value
        y : smaller value
    Process:
        System of linear equations:
            x + y = S
            x - y = D
        Solve:
            x = (S + D) / 2
            y = (S - D) / 2
    """
    x = (S + D) / 2
    y = (S - D) / 2
    return x, y

# ------------------------------------------------------------
# Plot function
# ------------------------------------------------------------
def plot_sum_diff(S, D):
    x, y = compute_values(S, D)

    fig, ax = plt.subplots(figsize=(8, 3))
    ax.set_xlim(0, S * 0.7)
    ax.set_ylim(-0.2, 1.2)
    ax.axis('off')

    # -------------------------------
    # Bars: Line segment diagram
    # -------------------------------
    ax.plot([0, x], [0.8, 0.8], lw=14, color="skyblue")
    ax.text(x + 20, 0.8, f"x = {int(x)}", va='center')

    ax.plot([0, y], [0.4, 0.4], lw=14, color="orange")
    ax.text(y + 20, 0.4, f"y = {int(y)}", va='center')

    # Difference bar
    ax.plot([y, x], [0.6, 0.6], lw=6, color="red")
    ax.text((x + y) / 2, 0.63, f"Difference = {D}", ha='center', color='red')

    # -------------------------------
    # System of Equations (drawn on figure)
    # -------------------------------
    eq_text = (
        r"System of Linear Equations:" "\n"
        r"$\;x + y = S$" "\n"
        r"$\;x - y = D$" "\n"
        fr"S = {S},  D = {D}" "\n"
        fr"$\Rightarrow x = {(S + D) / 2},\; y = {(S - D) / 2}$"
    )
    ax.text(0, -0.1, eq_text, fontsize=11, va='top')

    plt.title(f"Sum = {S}, Difference = {D}", fontsize=14)
    plt.show()

# ------------------------------------------------------------
# Widgets
# ------------------------------------------------------------
sum_slider = widgets.IntSlider(
    value=PARAM_INIT["SUM"], min=100, max=5000, step=50,
    description="Sum (S)", continuous_update=False
)

diff_slider = widgets.IntSlider(
    value=PARAM_INIT["DIFF"], min=0, max=2000, step=10,
    description="Diff (D)", continuous_update=False
)

out = Output()

def update_plot(*args):
    with out:
        out.clear_output(wait=True)
        plot_sum_diff(sum_slider.value, diff_slider.value)

sum_slider.observe(update_plot, names='value')
diff_slider.observe(update_plot, names='value')

# ------------------------------------------------------------
# Reset button
# ------------------------------------------------------------
reset_btn = Button(description="Reset", button_style="warning")

def reset_values(b):
    sum_slider.value = PARAM_INIT["SUM"]
    diff_slider.value = PARAM_INIT["DIFF"]

reset_btn.on_click(reset_values)

# ------------------------------------------------------------
# PNG save button
# ------------------------------------------------------------
save_btn = Button(description="Save PNG", button_style="success")

def save_png(b):
    ts = datetime.datetime.now().strftime("%Y%m%d_%H%M%S")
    fname = f"sumdiff_{ts}.png"
    S = sum_slider.value
    D = diff_slider.value

    x, y = compute_values(S, D)

    fig, ax = plt.subplots(figsize=(8, 3))
    ax.set_xlim(0, S * 0.7)
    ax.set_ylim(-0.2, 1.2)
    ax.axis('off')

    ax.plot([0, x], [0.8, 0.8], lw=14, color="skyblue")
    ax.plot([0, y], [0.4, 0.4], lw=14, color="orange")
    ax.plot([y, x], [0.6, 0.6], lw=6, color="red")

    ax.text(0, -0.1, f"x={(S + D)/2}, y={(S - D)/2}", fontsize=12)
    plt.title(f"Sum = {S}, Diff = {D}")
    plt.savefig(fname, dpi=200)
    plt.close()

    with out:
        print(f"Saved: {fname}")

save_btn.on_click(save_png)

# ------------------------------------------------------------
# Display UI
# ------------------------------------------------------------
display(VBox([
    HBox([sum_slider, diff_slider]),
    HBox([reset_btn, save_btn]),
    out
]))

# Initial plot
update_plot()