Question

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

Answer 1

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.