Struggling with System Solutions, Unit Vectors, and Matrices? Find Answers Here!

[ahmed dawod]

can anyone solve these
I tried too much to understand them but no way:

1- what values of a makes the system have no solution? one solution? many solutions?

x+2y-3z=4
3x-y+5z=2
4x+y+(a²-14)z=a+2

2- find all unit vectors in the plane determined by u=(3,0,1) , v =(1,-1,1)that are prependicular to the vector w=(1,2,0)

3- express the matrix
0 1 7 8
1 3 3 8
-2 -5 1 -8

in the form A = EFGR where E,F,G are elementary matrices and R is in Row-echelon form

4- show that the line
x=0 , y=t , z=t

a. lies in the plane 6x+4y-4z=0
b. is parallel to and below the plane 5x-3y+3z=1
c. is parallel to and above the plane 6x+2y-2z=3


please I have an exam and tried to solve these problems as I'm studying but no way

 

[lippyka]

1- For the system to have no solution, a2 - 14 must equal 0. For the system to have one solution, a2 - 14 must not equal 0. For the system to have many solutions, a2 - 14 must be a non-zero number. 2- The unit vectors perpendicular to w=(1,2,0) that lie in the plane determined by u=(3,0,1) and v=(1,-1,1) are (2/3, -1/3, 1/3) and (-1/3, 2/3, 1/3).3- A = EFGR, where E = 1 0 0 0 -1 0 1 00 1 0 0F = 1 0 0 00 1 0 0-2 0 1 0G = 1 0 0 00 1 0 00 0 1 -1R = 1 3 3 80 1 2 -10 0 0 04a- The line x=0, y=t, z=t lies in the plane 6x+4y-4z=0 if and only if 6(0)+4t-4t=0, which is true for any value of t. 4b- The line x=0, y=t, z=t is parallel to and below the plane 5x-3y+3z=1 if and only if 5(0)-3t+3t=1, which is true for any value of t. 4c- The line x=0, y=t, z=t is parallel to and above the plane 6x+2y-2z=3 if and only if 6(0)+2t-2t=3, which is true for any value of t.