Chose parameters h and k such that the system has a) a unique solution, b) many solutions, and c) no solution. x1 + 3x2 = 4 2x1 + kx2 = h

a) The system has a unique solution for and any value of , and we say the system is consisted

b) The system has infinite solutions for and

c) The system has no solution for and

Step-by-step explanation:

Since we need to base the solutions of the system on one of the independent terms (), the determinant method is not suitable and therefore we use the Gauss elimination method.

The first step is to write our system in the augmented matrix form:

The we can use the transformation , obtaining:

.

Now we can start the analysis:

from where we can see that only in the case of the value of can not be determined.

which means that the second equation is a linear combination of the first one. Therefore, we can solve the first equation to get as a function of o viceversa. Thus,   () is called a parameter since there are no constraints on what values they can take on.

if and the system has no solution. Again by substituting in the equation resulting from the last row:

which is false for all values of and since we have something that is not possible the system has no solution

the answer is c)8

step-by-step explanaction:

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