#A01 · Algebra & Number Theory · Lv.1View in 3D Mind-Map

Linear Equation & Balance Axiom

Al-Khwarizmi · 820 CE

ax+b=0  ⟹  x=−baax + b = 0 \implies x = -\frac{b}{a}
a≠0,f(x)=ax+ba \neq 0,\quad f(x) = ax + b

The foundational axiom of algebra (al-jabr): applying identical operations to both sides preserves equality and isolates the unknown x.

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Slope a
Intercept b
Target c
2x+(−6)=0  ⟹  x=0−(−6)2=3.0002x + (-6) = 0 \implies x = \frac{0 - (-6)}{2} = 3.000
Root x = (c-b)/a
3.0000
x-intercept (-b/a)
3.000
Slope Angle
63.4°

Prerequisite Path (1 STEP)

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STEP 1#A01Linear Equation & Balance Axiom

🚀 Next Unlocks (Derived Formulas) (10)

#A02Polynomial Expansion & Factoring#A03Laws of Exponents & Radicals#A04Quadratic Formula & Discriminant#A07Arithmetic & Geometric Series Sum#G09Euler's Polyhedron Formula & Topology#G10Parametric Bézier Curve Formula#C02Derivative Definition & Instantaneous Rate#L04Matrix Multiplication & Linear Transformation#L06Gaussian Elimination & Linear Systems (Ax = b)#P08Pearson Correlation & Linear Regression (OLS)

Geometric & Intuitive Insight

양팔 저울의 양쪽에 똑같은 무게를 덜어내어 미지의 추 x 하나만의 무게를 재는 과정입니다. 좌표평면에서는 직선 y = ax + b가 x축(y=0)과 만나는 절편을 뜻합니다.

Step-by-Step Derivation

  1. Step 1.1단계: 주어진 일차방정식 ax + b = 0 에서 상수항 b를 양변에서 뺍니다 → ax = -b
  2. Step 2.2단계: a ≠ 0 일 때 양변을 계수 a로 나누어 미지수 x를 고립시킵니다 → x = -b / a
  3. Step 3.3단계: 우변이 c인 일반형 ax + b = c 의 경우 x = (c - b) / a 로 즉시 확장됩니다.

AI & Coding Real-World Use

Single perceptron linear layer (y = wx + b) in AI and UI layout linear interpolation (Lerp)

#Linear Layer y=wx+b#UI Lerp Animation#Constraint Solver
const solveLinear = (a: number, b: number) => -b / a;

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