Question

Solve the system by Elimination Method፡ 3x+2y-z=-5

Answer 1

Answer:

Well, not much happens, but the process works the same as in any other case. In matrix terms, the equation is

[3−21]⎡⎣⎢xyz⎤⎦⎥=−5.

Now you are supposed to do row reduction to get an coefficient matrix in row-echelon form, but the current matrix is already in that form. According to the method, we should take x to be the basic variable, as it labels the single pivot column, and y and z to be the free variables. The method says to solve for the basic variables in terms of the free variables, which in this case would be

x=−53+23y−13z.

That makes the general solution

⎡⎣⎢xyz⎤⎦⎥=⎡⎣⎢−53+23y−13zyz⎤⎦⎥=⎡⎣⎢−5300⎤⎦⎥+y⎡⎣⎢2310⎤⎦⎥+z⎡⎣⎢−1301⎤⎦⎥,

or, by changing the parameters ( y=3s and z=3t ),

⎡⎣⎢xyz⎤⎦⎥=⎡⎣⎢−5300⎤⎦⎥+s⎡⎣⎢230⎤⎦⎥+t⎡⎣⎢−103⎤⎦⎥,

This solution is a plane obtained by translating along the vector [−5/300]⊤ the subspace spanned by [230]⊤ and [−103]⊤.