Question

Problem 4.1 Dependent equations? Given: x +2x2 2x+3x, 5 2xx +3x-2x 18 -x, + 3x2 + x3-4x,--6 x 3x2+ 5x 5x, 13 If the set of 4 equations is dependent or independent Find:

Answer 1

system Ax=b is

$\begin{pmatrix}1&2&-2&3\\ 2&-1&3&-2\\ -1&3&1&-4\\ 1&-3&5&-5\end{pmatrix}\begin{pmatrix}x_1\\ x_2\\ x_3\\ x_4\end{pmatrix}=\begin{pmatrix}5\\ 18\\ -6\\ 13\end{pmatrix}$

augmented matrix is

1	2	-2	3	5
2	-1	3	-2	18
-1	3	1	-4	-6
1	-3	5	-5	13

convert into Reduced Row Eschelon Form...

Add (-2 * row1) to row2

1	2	-2	3	5
0	-5	7	-8	8
-1	3	1	-4	-6
1	-3	5	-5	13

Add (1 * row1) to row3

1	2	-2	3	5
0	-5	7	-8	8
0	5	-1	-1	-1
1	-3	5	-5	13

Add (-1 * row1) to row4

1	2	-2	3	5
0	-5	7	-8	8
0	5	-1	-1	-1
0	-5	7	-8	8

Divide row2 by -5

1	2	-2	3	5
0	1	-7/5	8/5	-8/5
0	5	-1	-1	-1
0	-5	7	-8	8

Add (-5 * row2) to row3

1	2	-2	3	5
0	1	-7/5	8/5	-8/5
0	0	6	-9	7
0	-5	7	-8	8

Add (5 * row2) to row4

1	2	-2	3	5
0	1	-7/5	8/5	-8/5
0	0	6	-9	7
0	0	0	0	0

Divide row3 by 6

1	2	-2	3	5
0	1	-7/5	8/5	-8/5
0	0	1	-3/2	7/6
0	0	0	0	0

Add (7/5 * row3) to row2

1	2	-2	3	5
0	1	0	-1/2	1/30
0	0	1	-3/2	7/6
0	0	0	0	0

Add (2 * row3) to row1

1	2	0	0	22/3
0	1	0	-1/2	1/30
0	0	1	-3/2	7/6
0	0	0	0	0

Add (-2 * row2) to row1

1	0	0	1	109/15
0	1	0	-1/2	1/30
0	0	1	-3/2	7/6
0	0	0	0	0

reduced system is

$\begin{pmatrix}1&0&0&1\\ 0&1&0&-\frac{1}{2}\\ 0&0&1&-\frac{3}{2}\\ 0&0&0&0\end{pmatrix}\begin{pmatrix}x_1\\ x_2\\ x_3\\ x_4\end{pmatrix}=\begin{pmatrix}\frac{109}{15}\\ \frac{1}{30}\\ \frac{7}{6}\\ 0\end{pmatrix}$

there is no pivot entry at fourth column hence system has infinitely many solution

and these 4 equations are linearly dependent