🔗 Lesson 4: Systems of Equations

Solving multiple equations simultaneously — essential for multi-object physics problems

🎯 Learning Objectives

• Solve 2×2 systems using substitution and elimination
• Recognize when a physics problem requires a system of equations
• Interpret solutions: unique, infinite, or no solution

🔑 Key Concepts

When Do You Need a System?

Any time you have multiple unknowns and multiple independent relationships between them. Classic physics examples:

• Two-body collisions (momentum + energy → 2 equations, 2 unknowns)
• Tension in multiple ropes (force components → 2 equations, 2 unknowns)
• Circuit analysis (Kirchhoff's laws → multiple equations)
• Projectile motion (x and y components)

Method 1: Substitution

Best when one equation already has a variable isolated (or easily isolated).

1. Solve one equation for one variable
2. Substitute that expression into the other equation
3. Solve the resulting single-variable equation
4. Back-substitute to find the other variable

Method 2: Elimination (Addition/Subtraction)

Best when coefficients are set up for easy cancellation.

1. Multiply equation(s) to make coefficients of one variable opposites
2. Add equations to eliminate that variable
3. Solve for the remaining variable
4. Back-substitute

📐 Worked Examples

Example 1: Elastic Collision (Substitution)

A 2 kg ball moving at 4 m/s hits a stationary 3 kg ball. Find final velocities (elastic: KE conserved).

Equations: Momentum: 2(4) + 3(0) = 2v₁ + 3v₂ → 8 = 2v₁ + 3v₂

Energy: ½(2)(4)² = ½(2)v₁² + ½(3)v₂² → 16 = v₁² + 1.5v₂²

This is a quadratic system — we'll use the elastic collision shortcut: v₁f = (m₁-m₂)/(m₁+m₂)v₁i, v₂f = 2m₁/(m₁+m₂)v₁i

For practice, let's do a linear system instead...

Example 2: Tension Problem (Elimination)

A 10 kg mass hangs from two ropes at angles. T₁ at 30°, T₂ at 60°. Find tensions.

Horizontal: T₁cos30 = T₂cos60 → 0.866T₁ = 0.5T₂

Vertical: T₁sin30 + T₂sin60 = mg → 0.5T₁ + 0.866T₂ = 98

From Eq 1: T₂ = 1.732T₁
Substitute into Eq 2:
0.5T₁ + 0.866(1.732T₁) = 98
0.5T₁ + 1.5T₁ = 98
2T₁ = 98
T₁ = 49 N
T₂ = 1.732(49) = 84.9 N
            

Example 3: Two Unknown Forces (Substitution)

Two forces act on an object: F₁ at 0°, F₂ at 90°. Net force = 50 N at 36.9°. Find F₁, F₂.

x: F₁ = 50cos36.9° = 40

y: F₂ = 50sin36.9° = 30

Already separated! (Components naturally give a system.)

🏋️ Practice Problems

1. x + y = 10, x - y = 2 (solve by elimination)
2. 2x + 3y = 12, x = 2y (solve by substitution)
3. F₁ + F₂ = 100, 2F₁ - F₂ = 50 (physics: two forces)
4. m₁v₁ + m₂v₂ = 20, v₁ - v₂ = 5 (elastic collision setup, m₁=2, m₂=3)
5. T₁cosθ + T₂cosθ = 0, T₁sinθ + T₂sinθ = mg (symmetric ropes, θ=45°, m=10kg)
6. x + 2y = 8, 3x - y = 7
7. 0.5x + 0.2y = 1.4, 0.3x - 0.1y = 0.1 (decimals — clear them first!)
8. 2/p + 3/q = 5, 1/p - 1/q = 1 (substitute u=1/p, v=1/q)

✅ Answers

Click to reveal answers

1) x=6, y=4 2) x=4, y=2 3) F₁=50, F₂=50 4) v₁=7, v₂=2 5) T₁=T₂=69.3N

6) x=3, y=2.5 7) x=2, y=2 8) p=1, q=0.5

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