A linear equation has the form a₁y₁ + a₂y₂ + ... + a_ny_n = b. The numbers a_i are called the coefficients of the equation and the y_i are the variables. A solution to the equation is an ordered n-tuple of numbers (s₁,...,s_n) such that if the s_i are substituted for the y_i then the equation is true. For example, 3y₁ + 5y₂ = 0 is a linear equation. The coefficients are 3 and 5. The variables are y₁ and y₂. One solution to this equation is (5, -3) because "3 x 5 + 5 x (-3) = 0" is true. The set of all solutions to this equation is the set of all ordered pairs of the form (5a - 3b, -3a + 5b) where a and b are any numbers.

A system of linear equations is set of linear equations like so:

a₁₁ x y₁ + a...