1 ESERCIZI Sistemi di equazioni di primo grado PRACTICE WITH CLIL The Cramer s Rule A well-known rule for the solution of linear systems was elaborated by the Swiss mathematician Gabriel Cramer in the second half of XVIII century. The rule allows to reckon solutions for a linear system in the case of n equations in n unknowns. What we will do is to apply it to the simplest case of system of 2 equations in 2 unknowns. Let us see how it works. ax + by = c you extract a table, called «matrix , represented by the coefficients appearing in From the system { dx + ey = f the left-hand side of the two equations A = (a b) d e from this, you calculate the number D A = a e b d called «determinant of the matrix D following this pattern: | | DA = a b . d e Starting from matrix D, you work out two other matrices: one by replacing the constant coefficients to the first column of D and the other by replacing them to the second column of D. In both matrices, you calculate their determinants. Then: a c c b ; C = (d f ) B=( ) f e And the corresponding determinants: DB = c e b f ; DC = a f c d You will get the solutions by doing two divisions: DC D x = _B ; y = _ DA DA Its proof is rather complex if you consider the general case of n equations in n unknown systems, but for the case examined above it is a consequence of the addition and subtraction method. Let us apply the method to our case: To find x, you should delete y. You can get it by multiplying the first equation of the system by e, and the second by b then subtracting the second from the first. e ax + by = c b{dx + ey = f aex + bey = ce {bdx + bey = bf subtracting (ae bd)x + 0 = ce bf ce bf D x = _ = _B ae bd D A On the other hand, to find y, you should delete x. As before, you can get it by multiplying the first equation of the system by d, and the second by a then subtracting the second from the first. d ax + by = c a{dx + ey = f subtracting adx + bdy = cd 0 + (bd ae)y = cd af {adx + aey = af cd af y=_= bd ae af cd D C = ___________ = _ ae bd D A Exercises Solve the following systems using «Cramer s Rule 3x + 4y = 7 1. { [(5 ; 2)] x y=3 3x y = 1 2. [inconsistent] 1 _ {x 3 y = 2 3. 1 1 _ x + _y = 0 2 3 {3x + 2y = 0 [indeterminate] 37