⚖️ Lesson 2: Solving Linear Equations

Finding the unknown — the heart of algebraic problem-solving

🎯 Learning Objectives

🔑 Key Concepts

The Golden Rule: Keep the Balance

An equation is like a balance scale. Whatever you do to one side, you must do to the other.

Physics Insight: In physics, we often know the outcome (final velocity, total energy, net force) and need to find the cause (initial velocity, mass, acceleration). Solving equations is exactly this: working backward from what you know to find what you don't.

Inverse Operations

OperationInverse
Addition (+)Subtraction (−)
Subtraction (−)Addition (+)
Multiplication (×)Division (÷)
Division (÷)Multiplication (×)
Squaring (²)Square root (√)

General Strategy for Linear Equations

  1. Simplify both sides — distribute, combine like terms
  2. Move variable terms to one side — add/subtract to get all variables on one side
  3. Move constants to the other side
  4. Isolate the variable — divide by its coefficient
  5. Check — substitute back into original equation

📐 Worked Examples

Example 1: One-Step (Division)

Physics: F = ma. Find acceleration when F = 50 N, m = 10 kg.

50 = 10·a
50/10 = a          ← Divide both sides by 10
a = 5 m/s²

Example 2: Two-Step

Physics: v_f = v_i + at. Find time when v_f = 20 m/s, v_i = 5 m/s, a = 3 m/s².

20 = 5 + 3t
20 - 5 = 3t        ← Subtract 5 from both sides
15 = 3t
15/3 = t           ← Divide both sides by 3
t = 5 s

Example 3: Variables on Both Sides

Physics: Two objects with momenta p₁ = mv and p₂ = 2mv collide. Find v if total p = 30 kg·m/s and m = 2 kg.

mv + 2mv = 30
3mv = 30           ← Combine like terms
3(2)v = 30         ← Substitute m = 2
6v = 30
v = 5 m/s

Example 4: Fractions — Clear Denominators First!

Physics: 1/f = 1/do + 1/di (thin lens equation). Find di when f = 10 cm, do = 15 cm.

1/10 = 1/15 + 1/di
Multiply everything by 30 (LCM of 10, 15):
3 = 2 + 30/di
3 - 2 = 30/di
1 = 30/di
di = 30 cm
Pro Tip: When an equation has fractions, multiply every term by the least common denominator (LCD) to clear them. It's faster and prevents fraction errors.

🏋️ Practice Problems

  1. Solve for a: F = ma (F = 120 N, m = 8 kg)
  2. Solve for t: v = v₀ + at (v = 25, v₀ = 5, a = 4)
  3. Solve for x: x = x₀ + vt (x = 100, x₀ = 20, v = 10)
  4. Solve for m: p = mv (p = 50, v = 10)
  5. Solve for v: KE = ½mv² (KE = 200, m = 5)
  6. Solve for t: d = ½at² (d = 80, a = 4)
  7. 3x + 7 = 2x + 15
  8. 5(v - 2) = 3v + 10
  9. ½mv² = mgh (solve for v, m = 2, g = 10, h = 5)
  10. F₁ + F₂ = ma where F₁ = 2ma and F₂ = ma (solve for a, m = 4)

✅ Answers

Click to reveal answers

1) a = 15 m/s² 2) t = 5 s 3) x = 20 m 4) m = 5 kg 5) v = √80 = 8.94 m/s

6) t = √40 = 6.32 s 7) x = 8 8) v = 10 9) v = 10 m/s 10) a = 0 (forces cancel)

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