Question

solve and name each system question

Solve and name each system of equation 12. 4x + 3y =-13 -x + y = 5 13. 7x + 2y = 6 14x + 4y 12 14. 9x-5y 1 -18x + 10y = 1 15. 2x + y + z = 9 -x-y+ z 1 3x-y +z 9

Answer 1

solve by matrix method (all)

4x+3y=-13

-x+y=5

matrix form is

4	3	-13
-1	1	5

Divide row1 by 4

1	3/4	-13/4
-1	1	5

Add (1 * row1) to row2

1	3/4	-13/4
0	7/4	7/4

Divide row2 by 7/4

1	3/4	-13/4
0	1	1

Add (-3/4 * row2) to row1

1	0	-4
0	1	1

solution is $\large {\color{Red} (x,y)=(-4,1)}$

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.

7x+2y=6

14x+4y=12

matrix form is

7	2	6
14	4	12

Divide row1 by 7

1	2/7	6/7
14	4	12

Add (-14 * row1) to row2

1	2/7	6/7
0	0	0

last row is zero so system has infinitely many solution...because both line are same

$\large x=\frac{6}{7}-\frac{2}{7}y$...........and $\large y=y$

so general solution is

$\large {\color{Red} \begin{pmatrix}x\\ y\end{pmatrix}=\begin{pmatrix}\frac{6}{7}\\ 0\end{pmatrix}+\begin{pmatrix}-\frac{2}{7}\\ 1\end{pmatrix}y}$

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.

.

9x-5y=1

-18x+10y=1

matrix form is

9	-5	1
-18	10	1

Divide row1 by 9

1	-5/9	1/9
-18	10	1

Add (18 * row1) to row2

1	-5/9	1/9
0	0	3

Divide row2 by 3

1	-5/9	1/9
0	0	1

Add (-1/9 * row2) to row1

1	-5/9	0
0	0	1

from last row 0x+0y=1....wich is not possible so system has no solution beacause both line is parallel

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2x+y+z=9

-x-y+z=1

3z-y+z=9

matrix form is

2	1	1	9
-1	-1	1	1
3	-1	1	9

Divide row1 by 2

1	1/2	1/2	9/2
-1	-1	1	1
3	-1	1	9

Add (1 * row1) to row2

1	1/2	1/2	9/2
0	-1/2	3/2	11/2
3	-1	1	9

Add (-3 * row1) to row3

1	1/2	1/2	9/2
0	-1/2	3/2	11/2
0	-5/2	-1/2	-9/2

Divide row2 by -1/2

1	1/2	1/2	9/2
0	1	-3	-11
0	-5/2	-1/2	-9/2

Add (5/2 * row2) to row3

1	1/2	1/2	9/2
0	1	-3	-11
0	0	-8	-32

Divide row3 by -8

1	1/2	1/2	9/2
0	1	-3	-11
0	0	1	4

Add (3 * row3) to row2

1	1/2	1/2	9/2
0	1	0	1
0	0	1	4

Add (-1/2 * row3) to row1

1	1/2	0	5/2
0	1	0	1
0	0	1	4

Add (-1/2 * row2) to row1

1	0	0	2
0	1	0	1
0	0	1	4

solution is $\large {\color{Red} (x,y,z)=(2,1,4)}$