If `x ,ya n dz` are not all zero and connected by the equations `a_1x+b_1y+c_1z=0,a_2x+b_2y+c_2z=0,a n d(p_1+lambdaq_1)x+(p_2+lambdaq_2)+(p_3+lambdaq_3z=0)` , show that `lambda=-|a_1b_1c_1a_2b_2c_2p_1p_2p_3|-:|a_1b_1c_1a_2b_2c_2q_1q_2q_3|`

if x,y and z are not all zero and connected by the equations a_(1)x+b_(1)y+c_(1)z=0,a_(z)x+b_(2)y+c_(2)z=0 and (p_(1)+lambdaq_(1))x+(p_(2)+lambdaq_(2))y+(p_(3)+lambdaq_(3))z=0 show that lambda =-|{:(a_(1),,b_(1),,c_(1)),(a_(2) ,,b_(2),,c_(2)),(p_(1) ,, p_(2),,p_(3)):}|-:|{:(a_(1),,b_(1),,c_(1)),(a_(2) ,,b_(2),,c_(2)),(q_(1) ,, q_(2),,q_(3)):}|

For two linear equations , a_1x + b_1y + c_1=0 and a_2x + b_2y + c_2 = 0 , the condition (a_1)/(a_2)=(b_1)/(b_2)=(c_1)/(c_2) is for.

Consider the system of equations a_1x+b_1y+c_1z=0 a_2x+b_2y+c_2z=0 a_3x+b_3y+c_3z=0 if {:abs((a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)):}=0 , then the system has

Three linear equations a_1x+b_1y+c_1z=0, a_2x+b_2y+c_2z=0,a_3x+b_3y+c_3z=0 are consistent if (A) |(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|=0 (B) |(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|=-1 (C) a_1b_1c_1+a_2b_2c_2+a_3b_3c_3=0 (D) none of these

If a_1x+b_1y+c_1z=0, a_2x+b_2y+c_2z=0, a_3x+b_3y+c_3z=0 and |(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|=0 , then the given system then

If a_1x^3 + b_1x² + c_1x + d_1 = 0 and a_2x^3 + b_2x^2+ c_2x + d_2 = 0 have a pair of repeated roots common, then prove that |[3a_1,2b_1,c_1],[3a_2,2b_2,c_2],[a_2b_1-a_1b_2,c_1a_2-c_2a_1,d_1a_2-d_2a_1]|=0

Cosnsider the system of equation a_(1)x+b_(1)y+c_(1)z=0, a_(2)x+b_(2)y+c_(2)z=0, a_(3)x+b_(3)y+c_(3)z=0 if |{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=0 , then the system has

If (a_1)/(a_2) = (b_1)/(b_2) ne (c_1)/(c_2) , then the equations a_1x + b_1y + c_1 =0 and a_2x + b_2y + c_2=0 are _________. (consistent/inconsistent)

Consider the system of equations a_(1) x + b_(1) y + c_(1) z = 0 a_(2) x + b_(2) y + c_(2) z = 0 a_(3) x + b_(3) y + c_(3) z = 0 If |(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))| =0 , then the system has

Consider the system of linear equations 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 , Let us denote by Delta (a,b,c) the determinant |{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}| , if Delta (a,b,c) # 0, then the value of x in the unique solution of the above equations is