- #1

Specter

- 120

- 8

## Homework Statement

Determine whether the following system of equations has a single point of intersection. If so, find the point of intersection.

4x+y-9z=0

x+2y+3z=0

2x-3y-5=0

## Homework Equations

n

_{1}⋅(n

_{2}×n

_{3})

## The Attempt at a Solution

I have to pick a variable, use a pair of equations to eliminate the variable. Then I have to eliminate the same variable but with a different pair of equations. I tried doing this but I am not sure how correct it is.

n

_{1}=(1,1,-9)

n

_{2}=(1,2,3)

n

_{3}=(2,-3,0)

n

_{1}⋅(n

_{2}×n

_{3})

=(4,1,-9)⋅[(1,2,3)×(2,-3,0)]

=(4,1,-9)⋅(9,6,-7)

=105

105≠0, the normal vectors are not coplanar so there is a single point of intersection.4x+y-9z=0 [1]

x+2y+3z=0 [2]

2x-3y-5=0 [3]

4x+y-9z=0 [1]

4x+8y+12z=0 [4] Eqn [2] x 4 to eliminate x.

Subtract and the new equation is 7y+3z=0 [5].

Use a different pair of equations to eliminate x. I used equations 1 and 3.

4x+y-9z=0 [1]

4x-6z-10=0 [6] (Eqn 3x2)

Subtract and the new equation is -5y-9z-10=0 [7]

The new system with 2 eqns and 2 variables:

7y+3z=0

-5y-9z-10=0

If this is correct, I sort of know where to go from here. I would solve for y and z.