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
