Total number of solutions to the system of linear equations x + y + 0.z = 0, X – y + 0.z = 0, X + 4y + 0.z = 0 is infinity.
Total number of solutions to the system of linear equations x + y + 0.z = 0, X – y + 0.z = 0, X + 4y + 0.z = 0 is infinity.
The correct option is A True
Once you express the given equations in the form of matrices the determinant D = 0 and D1 = D2 = D3 = 0. So according to Cramer's rule there exists either infinite number of solutions or no solutions at all. Here is when u can see the advantage of relating this to 3d geometry. If you visualize the 3 planes the planes x + y = 0, X – y = 0 intersect along z axis and the plane x + 4y = 0 also passes through the z axis. Therefore they intersect in a straight line along z axis I.e infinite solutions.
Alternate explanation: once we know that there exists no or infinite solutions we can see x = 0, y = 0 satisfies all the 3 equations irrespective of the value of z therefore any triplet of the for (x,y,z) = (0,0,z) satisfies these equations and the z can take any real value therefore infinite solutions
True
Once you express the given equations in the form of matrices the determinant D = 0 and D1 = D2 = D3 = 0. So according to Cramer's rule there exists either infinite number of solutions or no solutions at all. Here is when u can see the advantage of relating this to 3d geometry. If you visualize the 3 planes the planes x + y = 0, X – y = 0 intersect along z axis and the plane x + 4y = 0 also passes through the z axis. Therefore they intersect in a straight line along z axis I.e infinite solutions.
Alternate explanation: once we know that there exists no or infinite solutions we can see x = 0, y = 0 satisfies all the 3 equations irrespective of the value of z therefore any triplet of the for (x,y,z) = (0,0,z) satisfies these equations and the z can take any real value therefore infinite solutions
