Where are you stuck? What have you tried?
Consider the following set of linear equations
(h = lamda)
(h-3)x + y =0
x + (h-4)y =0
i)for which value(s) of the parameter h will the system have nontrivial solutions?
ii)find the solution space for the case when there are many solutions?
b)Consider the following set of linear equations:
a(x1) - 2(x2) = k
2(x1) - 2(x2) = 2
i)Find the values of a and k such that the system has infinite,one and none solutions
ii)Find the solution and the solution space for the cases when the system has one and infinitely many solutions respectively
Can you show as much working as possible
For the first part, remember that a trivial solution is one in which the only way to make the solution to the homogeneous equation (which you have) is if both X and Y are equal to zero. We see that if H is equal to 3, then Y is equal to 0 (from the first equation). And if Y is equal to 0 in the first equation, the so too much X be equal to zero in the second equation. The same is true of 4 in the second equation forcing X to be zero in the second, and Y to be zero in the first. Therefore, H can be any value except for 3 and 4.
For the second part, for the system to have infinitely many solutions, then A must equal 2 and K must equal 2 as well (as then you will have equivalent statements). For the case of having one solution, solve for X1 and X2 using substitution (simply solve one of the equations for one of the variables, and then plug it into the remaining equation).
I will leave the solution space for you to solve.