Problem A

(eq1): x + 2y + 3z = 0

(eq2): x + y + z = 0

(eq3): 3x + 2y + z = 0

eliminating x:

(eq2) - (eq1)

(eq4): y + 2z = 0

3(eq1) - (eq3)

(eq5): 4y + 8z = 0

eliminating y:

4(eq4) - eq(5)

0 = 0

The system is consistent, and does not have a unique solution.

The planes intersect like the pages of an open book along the spine.

Equations describing the plane are:

z = t

y = -2t

x + y + z = 0

x = t

So (x,y,z) = (t, -2t, t) where t is any real number.

Notice that we could see easily that (x,y,z) = (0,0,0) was one possible solution; but it turned out there were many others as well.

Problem B

(eq1): x - 2y - 2z = 0

(eq2): 3x + 2y + z = 2

(eq3): x + 14y + 12z = 4

eliminating x:

3(eq2) - (eq1)

(eq4): 8y + 7z = 2

(eq3) - (eq1)

(eq5): 16y + 14z = 4

eliminating y:

(eq5) - 2(eq4)

0 = 0

The system is consistent, and does not have a unique solution. It is represented by three planes intersecting along a common line.

Equations describing the plane are:

z = t

8y = 2 - 7t

y = 0.25 - 0.875t

x - 2y - 2z = 0

x = 2y + 2z

x = 0.5 - 3.75t

So (x,y,z) = (0.5 - 3.75t, 0.25 - 0.875t , t) where t is any real number represents the line.

Problem C

(eq1): 2x + 0.5y + z = 2

(eq2): 2x + 2y + 2.4z = 3

(eq3): 1.4x + 1.1y + 1.4z = 1.9

eliminating x:

(eq2) - (eq1)

(eq4): 1.5y + 1.4z = 1

(eq3) - 0.7(eq1)

(eq5): 0.75y + 0.7z = 0.5

eliminating y:

(eq4) - 2(eq5):

0 = 0

The system is consistent, and does not have a unique solution.

The planes intersect like the pages of an open book along the spine.

Equations describing in intersection of the planes above:

z = t

1.5y = 1- 1.4t

y = 0.6667 - 0.9333t

x = 0.8333 - 0.2667t

Problem D

The first two equations represent the same plane.

We can effectively ignore (eq2).

(eq1): x + y + z = 1

(eq3): 3x + 7y + 11z = 101

eliminating x:

(eq3) - 3(eq1)

4y + 8z = 98

y + 2z = 24.5

Let z = t a real number.

Then y = 24.5 - 2t

Substituting into (eq1)

x = 1 - y - z

x = -23.5 - 3t

So (x,y,z) = (-23.5 - 3t , 24.5 - 2t , t) is the line of solutions.