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Find a linear function whose graph is the plane that intersects the xy-plane along the line y= x+7 and contains the point (-4,-4,-21).

Find a linear function whose graph is the plane that intersects the xy-plane along the line y= x+7 and contains the point (-4,-4,-21).
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1 Answer

One way to solve this kind of problem is to use a description of lines drawn from formal linear algebra.
 
The line (which will be referred to as Line1):  y = x +7   can be described by the ordered triplet (x,y,z)
  (0, 7, 0) + α (1 , 1 ,0) =   ( α , α +7, 0)      The parameter α can take on all real values.  It is the parameter that moves a point along the line.
 
  Line 1 is along the 45 degree direction in the x,y plane.   The vector  (1, 1, 0) above generates a 45 degree line in the x,y plane.   A similar vector, namely (1, -1, c) will generate a line in the 3-space that will either be skew to Line1 or intersect line Line1 at right angles.  This is because the dot product between (1,1,0) and (1, -1, c) is zero.
 
To get the intersection case for Line2   we can write for Line2
 
    ( α , α +7, 0) + β (1, -1, c)     The parameter β plays a role for Line2, similar to that of α for Line1
    This expression for Line2 is actually a family of lines indexed by α.  The desired line is one member of this family.
 
   The next step is to set  ( α , α +7, 0) + β (1, -1, c)   = (-4, -4, -21)      This can be viewed as a set of three equations for three unknowns (α, β, c ).  The equations are:
 
       α + β  =  -4
       α + 7 - β  = -4
       β c         =  -21
 
The first two equations can be solved  to get  α = -15/2    β = 7/2
 
The third equation then gives   c = -6
 
  Substituting for α and c gives for Line2      (-15/2 + β , -1/2 - β , -6 β)
  For  β = 7/2  this form generates (-4 , -4, -21)  as required.   Also with β =0   we recover ( α , α +7, 0)   when α =-15/2  .   These two facts satisfy the requirements for Line2:  It passes through (-4,-4,-21) and lies in a plane intersecting the x,y plane along the line  y =  x +7 .
 
From the viewpoint of linear algebra,  the form (-15/2 + β , -1/2 - β , -6 β)  is a complete answer.   It can be converted to alternative forms.   For example eliminating β in favor of z  results in the set of two equations
 
       x = -15/2   -  z/6     and
       y = -1/2  + z/6