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Solving Linear Equations

What

A linear equation is an equation where all variables appear to the power 1 (no exponents, products, or functions like sin or log).

A system of linear equations is a set of linear equations with the same variables.

General form (for one equation in nn variables): \(a_1x_1 + a_2x_2 + \dots + a_nx_n = b\)

It looks like this: \(ax+b=c\)

  • \(x\) is the unknown (what we want to find).
  • \(a,b,c\) are numbers (called constants).
  • The graph of a linear equation is always a straight line.

How

Number of Variables vs Number of Equations

  • Let \(n\) = number of variables
  • Let \(m\) = number of equations

Cases

  1. Exactly determined

    If \(m = n\):

    • The system might have a unique solution.
    • Depends on whether equations are independent.
    • A unique solution means the lines intersect at one point.

  2. Underdetermined

    If \(m < n\):

    • Fewer equations than unknowns
    • Infinite solutions (if consistent) or none
    • Example: 2 variables, 1 equation → line in 2D

  3. Overdetermined

    If \(m > n\):

    • More equations than unknowns
    • May have no solution (inconsistent)
    • If consistent, still can have a unique solution (e.g. in least squares problems)

Why

Understanding the structure of linear systems is key to:

  • Linear algebra
  • Data science (e.g. regression)
  • Physics (e.g. forces in equilibrium)
  • Engineering (e.g. circuit analysis)
  • Optimization

Linear systems are the building blocks for more complex models in applied math and machine learning.