Drawing a plane on a cartesian plane using gnuplot -

i'm trying reproduce figure i've found on linear algebra book using gnuplot. original image

you can see intersection between 2 planes described 2 equations:

• 2u + v + w = 5
• 4u - 6v = -2.

i suppose in order plot first equation using gnuplot have transform in form:

splot 5 - 2*x - y

where u -> x; v -> y , w -> z free variable. result different expected. clue?

the approach outline makes sense, however, results may far expect.
propose draw single lines, using arrow function in gnuplot.
example generate plot similar 1 showed (only 1 plane, though):

set term gif set output "demo_plane.gif"  # define axis limits: xmax =   6.5              xmin =  -1.5 ymax =   8.5                  ymin =  -1.5 zmax =   5.5 zmin =  -0.5                   set xrange [xmin:xmax] set yrange [ymin:ymax] set zrange [zmin:zmax]  # remove original axis unset border unset xtics unset ytics unset ztics  # define data points: x1 =  3.0 y1 = -1.0 z1 =  0.0  x2 = -1.0 y2 =  7.0 z2 =  0.0  x3 = -3.0 y3 =  7.0 z3 =  4.0  x4 =  1.0 y4 = -1.0 z4 =  4.0  # define 'arrow' without head: set arrow 1 x1,y1,z1 \               x2,y2,z2 nohead    set arrow 2 x2,y2,z2 \               x3,y3,z3 nohead    set arrow 3 x3,y3,z3 \               x4,y4,z4 nohead    set arrow 4 x4,y4,z4 \               x1,y1,z1 nohead    # draw new axis manually (again, using arrow): set arrow 5 0,0,0 \               6,0,0     set arrow 6 0,0,0 \               0,6,0     set arrow 7 0,0,0 \               0,0,5   # annotate axis labels: set label "u" @ 6.25,0,0 set label "v" @ 0,6.25,0 set label "w" @ 0,0,5.25  # plot not show when empty, include dummy plot command: set parametric  splot x1, y1, z1 not 

with little rotation figure this: