Answer:
a) The set of solutions is
y b) the set of solutions is
.
Step-by-step explanation:
a) Let's first find the echelon form of the matrix
.
- We add
from row 1 to row 2 and we obtain the matrix ![\left[\begin{array}{ccc}5&2&6\\0&\frac{9}{5} &\frac{27}{5}\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D5%262%266%5C%5C0%26%5Cfrac%7B9%7D%7B5%7D%20%26%5Cfrac%7B27%7D%7B5%7D%5Cend%7Barray%7D%5Cright%5D)
- From the previous matrix, we multiply row 1 by
and the row 2 by
and we obtain the matrix
. This matrix is the echelon form of the initial matrix.
The system has a free variable (x3).
- 0=x1+
x2+
x3=
x1+
(-3x3)+
x3=
x1-
x3+
x3
then x1=0.
The system has infinite solutions of the form (x1,x2,x3)=(0,-3x3,x3), where x3 is a real number.
b) Let's first find the echelon form of the aumented matrix
.
- To row 2 we subtract row 1 and we obtain the matrix
![\left[\begin{array}{ccccc}1&-2&1&-4&1\\0&5&6&6&1\\0&-12&-11&-16&5\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccccc%7D1%26-2%261%26-4%261%5C%5C0%265%266%266%261%5C%5C0%26-12%26-11%26-16%265%5Cend%7Barray%7D%5Cright%5D)
- From the previous matrix, we add to row 3,
of row 2 and we obtain the matrix
.
- From the previous matrix, we multiply row 2 by
and the row 3 by
and we obtain the matrix
. This matrix is the echelon form of the initial matrix.
The system has a free variable (x4).
- x3-
x4=
, then x3=
+
x3+
x4=
, x2+
(
+
x4=
, then
x2=
x4.
- x1-2x2+x3-4x4=1, x1+
+
x4+
+
x4-4x4=1, then x1=![1-\frac{119}{17}=-6](https://tex.z-dn.net/?f=1-%5Cfrac%7B119%7D%7B17%7D%3D-6)
The system has infinite solutions of the form (x1,x2,x3,x4)=(-6,
x4,
+
x4,x4), where x4 is a real number.