in The double subscripting on the coefficients aij the first coefficient indicates which unknown it multiplies

in The double subscripting on the coefficients aij the second coefficient indicates the equation in which the coefficient occurs

a12

Number of solutions that can occur in linear system

consider that we have 2 equations and the lines of the 2 equations are in parallel and there’s no intersection. So we have ….

consider that we have 2 equations and the lines of the 2 equations intersect at only one point . So we have ….

consider that we have 2 equations and the lines of the 2 equations coincide. So we have ….

many points of intersection

the points on the common line

if the system has at least one solution we call it as

if the system has no solutions we call it as

if we have 3 planes and all of these planes are in parallel we have

if we have 3 planes and 2 of these planes are in parallel, we have

if we have 3 planes and there’s no common intersection, we have

if we have 3 planes and they all intersect each other but there’s no common intersection, we have

if we have 3 planes, 2 of them are coincident and they are parallel to the third, we have

if we have 3 planes and the intersection of them is one point, we have

if we have 3 planes and the intersection of them is a line, we have

if we have 3 planes and they are all coincident, we have

if we have 3 planes and their intersection is a plane, we have

Every system of linear equations has zero, one or infinitely many solutions. There are no other possibilities