When the dependent variable Y is influenced by more than one independent variable, we use Multiple Linear Regression.
MLR Model
Ŷ = a + b₁X₁ + b₂X₂ + ... + bkXk
Where:
Ŷ = predicted value of dependent variable
a = intercept (constant)
b₁, b₂, ..., bk = partial regression coefficients
X₁, X₂, ..., Xk = independent variables
For two independent variables (most common in exam):
Ŷ = a + b₁X₁ + b₂X₂
In MLR, b₁ and b₂ are called partial regression coefficients. b₁ represents the change in Y per unit change in X₁, keeping X₂ constant. Similarly for b₂.
Normal Equations (MLR with two predictors)
ΣY = na + b₁ΣX₁ + b₂ΣX₂
ΣX₁Y = aΣX₁ + b₁ΣX₁² + b₂ΣX₁X₂
ΣX₂Y = aΣX₂ + b₁ΣX₁X₂ + b₂ΣX₂²
Solve these three equations simultaneously to find a, b₁, and b₂.