Question

Solve and Graph the System of Linear Inequalities below:

x ≥ 0
y ≥ 0
y + x ≤ 6
y + 3x ≤ 12

Please show ALL working as much as possible and include in your answer a coordinate plane with all the included graphs (for the above equations) and the area/region of solutions appropriately shaded.

Answer 1

The below graph with portion enclosed by red lines is the solution to the given equations.

Explanation:

`x\geq  0` means the portion to the right of
`y-axis`

`y\geq  0` means the portion up of the
`x-axis`

For
`y + x \leq 6` , we find
`x-interecpt` and
`y-intercept`

For
`x-interecpt` we substitute y=0

`y+x=6\\0+x=6\\x=6`,thus pont is
`(6,0)`

Similary
`y-intercept` , we substitute x=0

`y + x= 6\\y+0=6\\y=6`, thus point is
`(0,6)`

We connect these these two points and consider the below area as it contains less than inequality

For
`y + 3x\leq 12` we again find the
`x-intercept` and
`y-intercept`

For
`x-interecpt` we substitute y=0

`y+3x=12\\0+3x=12\\`

divide both side by
`3` , we get

`x=4`, thus point is
`(4,0)`

Similary
`y-intercept` , we substitute x=0

`y+3x=12\\y+0 =12\\y=12`

Thus point is
`(12,0)` ,

Now connect these two points and consider the area below the line as it contains less than inequality.

Finally the area enclosed by all these lines is the solution to the given equations, that is shown below.