Solve: 3x + 7 = 22

Step 1: Subtract 7 from both sides
  3x + 7 − 7 = 22 − 7
  3x = 15

Step 2: Divide both sides by 3
  3x/3 = 15/3
  x = 5

Verify: 3(5) + 7 = 15 + 7 = 22 ✓

General formula:
  ax + b = c
  x = (c − b) / a

Special cases:
  a = 0, b = c → identity (all x)
  a = 0, b ≠ c → no solution
  One variable, one equation → unique solution

Linear equations always have exactly one solution (when a≠0). The solution process is reversible: each step applies an inverse operation (addition↔subtraction, multiplication↔division) to both sides.

The Quadratic Formula:

        −b ± √(b² − 4ac)
  x = ─────────────────────
              2a

Discriminant Δ = b² − 4ac
  Δ > 0 → Two distinct real roots
  Δ = 0 → One repeated real root
  Δ < 0 → Two complex conjugate roots

Example: 2x² − 7x + 3 = 0
  a=2, b=−7, c=3
  Δ = 49 − 24 = 25
  x = (7 ± 5) / 4
  x₁ = 12/4 = 3
  x₂ = 2/4 = 0.5

Alternative methods:
  • Factoring: (2x−1)(x−3) = 0
  • Completing the square
  • Graphing (x-intercepts)

The quadratic formula was essentially known to Babylonians (~2000 BC) and formally derived by al-Khwarizmi (820 AD). It works for ALL quadratics — factoring only works when roots are rational.

System of 2 equations:
  2x + 3y = 12
  4x − y  = 5

Method 1: Substitution
  From eq.2: y = 4x − 5
  Substitute into eq.1:
    2x + 3(4x − 5) = 12
    2x + 12x − 15 = 12
    14x = 27
    x = 27/14 ≈ 1.929
    y = 4(27/14) − 5 = 38/14 ≈ 2.714

Method 2: Elimination
  Multiply eq.2 by 3:
    12x − 3y = 15
  Add to eq.1:
    2x + 3y = 12
    14x = 27 → same result

Method 3: Cramer's Rule
  |A| = 2(−1)−3(4) = −14
  x = |Ax|/|A| = (12(−1)−3(5))/(−14)
  x = −27/(−14) = 27/14 ✓

For 2×2 systems, any method works. For larger systems (3+ variables), Gaussian elimination or matrix methods (LU decomposition) are more systematic. Cramer's Rule is elegant but computationally expensive for large systems.

Iterative root-finding for f(x) = 0:

  x_{n+1} = x_n − f(x_n) / f'(x_n)

Example: Find √2 (solve x² − 2 = 0)
  f(x) = x² − 2, f'(x) = 2x
  Start: x₀ = 1.5

  x₁ = 1.5 − (2.25−2)/(3)
     = 1.5 − 0.0833 = 1.41667

  x₂ = 1.41667 − (2.00694−2)/(2.83333)
     = 1.41667 − 0.00245 = 1.41422

  x₃ = 1.41421 (6 correct digits!)

Convergence: Quadratic
  Digits of accuracy roughly double
  each iteration near the root.

Warnings:
  • Needs good initial guess
  • Fails if f'(x_n) = 0
  • May diverge for poor starts
  • Multiple roots → different starts

Newton-Raphson converges extremely fast (quadratic convergence) when it works. Most scientific calculators and computer algebra systems use it internally. For guaranteed convergence, bisection method is slower but always works on continuous functions.