Page:Elementary algebra (1896).djvu/462

444 Rh APPLICATION TO THE SOLUTION OF SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE.

552. The properties of determinants may be usefully employed in solving simultaneous linear equations. Let the equations be

a_1 x+b_1 y+c_1 z+d_1=0,

a_2 x+b_2 y+c_2 z+d_2=0,

a_3 x+b_3 y+c_3 z+d_3=0;

multiply them by A_1, -A_2, A_3 respectively and add the results, A_1, A_2, A_3, being minors of a_1, a_2, a_3 in the determinant

D= a_1 b_1 c_1 a_2 b_2 c_2 a_3 b_3 c_3

The coefficients of y and z vanish in virtue of the relations proved in Art. 548, and we obtain

(a_1 A_1 - a_2 A_2 + a_3 A_3)y+ (d_1 A_1 - d_2 A_2 + d_3 A_3) = 0.

Similarly we may show that

(b_1 B_1 - b_2 B_2 + b_3 B_3 )y + (d_1 B_1 - d_2 B_2 + d_3 B_3 ) = 0,

and = (c_1 C_1 - c_2 C_2  + c_3 C_3 )z + (d_1 C_1 - d_2 C_2 + d_3 C_3 ) = 0.

Now a_1 A_1 - a_2 A_2 + a_3 A_3 = -(b_1 B_1 - b_2 B_2 + b_3 B_3 ) = (c_1 C_1 - c_2 C_2  + c_3 C_3) = D;

hence the solution may be written

or more symmetrically