Consider the system w+ 3x + 2y -3z = 0 -3w -6x + 3y -4z = 0 4w -5x -3y + 3z = 0 Explain, without solving, how many solutions the system has and then find the general solution of the system and...

Consider the system

w+ 3x + 2y -3z = 0

-3w -6x + 3y -4z = 0

4w -5x -3y + 3z = 0

Explain, without solving, how many solutions the system has and then find the general solution of the system and express it in vector form.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The system of equations given consists of 3 equations each with 4 variables, w, x, y and z. As the total number of equations is less than the number of variables, the system of equations does not have an independent solution. Each of the four variables can be assigned a value and the corresponding values of the other three determined.

The system of equations has an infinite number of solutions.

rakesh05's profile pic

rakesh05 | High School Teacher | (Level 1) Assistant Educator

Posted on

As the given homogeneous system three equations in four variables. So, certainly at  least one variable become the free variable. And hence the system possess infinitely many solutions. And the values of other variable will come in terms of the free variable.

Now we put the given equation in matrix form as

[[1,3,2,-3],[-3,-6,3,-4],[4,-5,-3,3]]~[[1,3,2,-3],[0,1,3,-13/3],[0,0,40,-176/3]]

we can write 40y-(176/3)z=0

or,                  y=(176/120)z

                     x=(-1/15)z

                     w=(1/5)z

                  Take z=k

                    W=(1/5)k

                     x=(-1/15)k

                    y=(176/120)k  will serve the general solution.

In vector form the general solution is [(1/5)k, (-1/15)k, (176/120)k, k].

 

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