How do you graph the system #x - 4y >= -4# and #3x + y <= 6#?

1 Answer
Maggie
Feb 22, 2016

1) Graph the line #y=1/4 x + 1#,
it has a slope of 1/4 and a y intercept of 1.
2) The region #x-4y>=-4# (or #y<=1/4 x + 1#) is the area below this line and the line itself, shade/hatch this region.

3) Graph the line #y=-3x+6#,
it has a slope of -3 and a y intercept of 6.
4) The region #3x+y<=6# (or #y<=-3x+6#) is the area below this line and the line itself, shade/hatch this region a different colour/pattern from the other region.

5) The SYSTEM, is the set of x and y values the satisfy both expressions. This is intersection of both regions. Whatever both shades occur is the graph of the system.

Explanation:

Consider the region defined by #x-4y>=-4#.
The edge of the region is defined by the equation #x-4y=-4#.
This need to be put in standard form.

Start with,
#x-4y>=-4#
Subtract x from both sides.
#x-4y-x>=-4-x#
Producing,
#-4y>=-4-x#.

Divide both side by -4 (remember to flip the inequality)
#{-4y}/-4<={-4-x}/-4#.
We have
#y<=1+x/4# or #y<=1/4 x + 1#.
The edge is the line y=1/4 x + 1 and the region the area below this including the line.

Consider the region defined by #3x+y<=6#.
The edge of the region is defined by the equation #3x+y=6#.
This need to be put in standard form.

Start with,
#3x+y<=6#
Subtract 3x from both sides.
#3x+y-3x<=6-3x#
Producing,
#y<=6-3x#
or
#y<=-3x+6#

The edge is the line y=-3x+6 and the region the area below this including the line.

The SYSTEM, is the set of x and y values the satisfy both expressions. This is intersection of both regions.

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