x+6y+3z=4,2x+y+2z=3,3x-2y+z=0

Question

There is 3 different parts and I need help setting it up.

Answer 1

x+6y+3z=4
2x+y+2z=3
3x-2y+z=0

Let’s convert each equation to a row of coefficients + constant:

x, y, z, c
R1 1, 6, 3, 4
R2 2, 1, 2, 3
R3 3,-2, 1, 0

We can do anything to the rows that we can do to the equations.

R1 > 6*R1
R2 > 3*R2
R3 > 2*R3

R1 6,36,18,24
R2 6, 3, 6, 9
R3 6,-4, 2, 0

R1 > R1 - R3
R2 > R2 - R3

R1 0,40,16,24
R2 0, 7, 4, 9
R3 6,-4, 2, 0

R1 > R1/8
R3 > R3/2

R1 0, 5, 2, 3
R2 0, 7, 4, 9
R3 3,-2, 1, 0

R1 > 7*R1
R2 > 5*R2

R1 0, 35, 14, 21
R2 0, 35, 20, 45
R3 3,-2, 1, 0

R2 > R2 - R1

R1 0,35,14,21
R2 0, 0, 6,24
R3 3,-2, 1, 0

R1 > R1/7
R2 > R2/6

R1 0,5, 2, 3
R2 0, 0, 1,4
R3 3,-2, 1, 0

R1 > R1 - 2*R2
R3 > R3 - R2

R1 0, 5, 0,-5
R2 0, 0, 1, 4
R3 3,-2, 0,-4

R1 > R1/5

R1 0, 1, 0,-1
R2 0, 0, 1, 4
R3 3,-2, 0,-4

R3 > R3 + 2*R1

R1 0, 1, 0,-1
R2 0, 0, 1, 4
R3 3, 0, 0,-6

R3 > R3/3

R1 0, 1, 0,-1
R2 0, 0, 1, 4
R3 1, 0, 0,-2

(x,y,z)=(-2,-1,4)

check:

(-2)+6(-1)+3(4)=?4
-2-6+12=4 √

2(-2)+(-1)+2(4)=?3
-4-1+8=3 √

3(-2)-2(-1)+(4)=?0
—6+2+(4)=0 √