29. Homogeneous variable

Another method for the solution of non-homogeneous, linear equations involves the introduction of homogenous variables. In (26.1), set

x₁ = t₁/t_1n+1, x₂ = t₂/t_1n+1, ··· , x₁ = t_n/t_1n+1

and multiply by t_1n+1. Then (26.1) becomes the homogeneous system

a₁₁t₁ + ··· + a_1nt_n - c₁t_n+1 = 0,
····························································(29.1)
a_p1t₁ + ··· + a_pnt_n - c_pt_n+1 = 0.

Every solution of (29.1) in which t_n+1 0 yields a solution of (26.1). However, you must find out whether (29.1) has solutions with t_n+1 0.

According to Theorem 14, the number of linearly independent solutions of (29.1) is n + 1 - r₁. According to the same Theorem,

a₁₁t₁ + ··· + a_1nt_n = 0,
····························································(29.2)
a_p1t₁ + ··· + a_pnt_n = 0

has exactly n - r linearly independent solutions. Every solution of (29.1) with t_n+1=0 yields a solution of (29.2) and every solution of (29.2) a solution of (29.1) with t_n+1 = 0, whence (29.1) has exactly then solutions with t_n+1 0, if

n + 1 - r₁ > n - r

or r₁ < r + 1. On the other hand, certainly r₁ = r or r₁ = r + 1, so that is r₁<r+ 1 is equivalent to r₁ = r. Hence you have arrived again at the statement of Theorem 15.

last next