Question

) The points (1, 0, 3), (1, 1, 1), and (−2, −1, 2) lie on a unique plane a1x1 + a2x2 + a3x3 = b. Using your previous answers, find an equation for this plane. (Hint: think about the relationship between the previous system and the one you would need to solve in this question.)

Answer 1

2x₁ - x₂ + x₃ = 2

Question in full

(a) Find the general solution for the following system of linear equations:

z2 + 3z3 − z4 = 0

−z1 − z2 − z3 + z4 = 0

−2z1 − 4z2 + 4z3 − 2z4 = 0

(b) Give an example of a solution to the previous system of linear equations.

(c) The points (0, 1, 3), (1, 1, 1), and (−1, −2, 2) lie on a unique plane in R 3 , defined by an equation of the form a1x1 + a2x2 + a3x3 = b. Using your previous answers, find an equation for this plane. (Hint: think about the relationship between the previous system and the one you would need to solve in this question.)

Explanation:

The augmented matrix is
`\left[\begin{array}{ccc}1&2&3&0\\4&5&6&0\\7&8&9&0\end{array}\right]`

The echelon form

The general solution and the sample solution

as well as the equation of the plane

are given in the attached solution below

Answer 2

x1-2x2-x3= -2

Explanation:

Substitute the three points to get three equations:

a1+0a2+ 3a3=d --------------eq i

a1+a2+a3=d----------------eq ii

-2a1-a2+2a3=d---------------eq iii

subtract eq i from eq ii

a2-2a3=0 ----------- eq iv

now add eq iv to eq iii

-2a1=d ------- eq v

put eq v in eq i

a1 + 3a3=-2a1

a3= -a1 ----------- eq vi

put eq vi in eq iv

a2-2(-a1)=0

a2= -2a1 ------ eq vii

put eq v, eq vi and eq vii in the equation of plane

a1x1+ a2x2+a3x3=d

a1x1 -2a1x2-a1x3=-2a1

x1-2x2-x3= -2