Step 1:

The first inequality is $ x+2y \leq 10$---------(1)

Consider the equation $x+2y =10$

The points $(10,0) $ and $ (0,5)$ satisfy the equation .

The graph of the equation is drawn as shown .

The line divides the xy plane into two half planes.

Consider the point$(0,0)$

We see that, $ 0+ 2(0) \leq 10$

$=> 0 \leq 10$ is true.

Hence the inequality (1) represents the region below the line $ x+2y =10$ containing the point (0,0) (including the line )

Step 2:

The second inequality is $x+y \leq 1$ -----(2)

Consider the equation $x+y =1$

We see that $(1,0)$ and $(0,1)$ satisfy the equation .

The graph of the equation is drawn as shown.

The line divides the XY plane into two regions ( half plane)

Consider the point (0,0)

We see that $ 0+0 \geq 1$

$ => 0 \leq 1$ is false

Thus the inequality (2) is represented by the region above the line $x+y=1$ not including the point (0,0) (including the line y=2x)

Step 3 :

The third inequality is $x-y \leq 0$ -----(3)

Consider the equation $x-y =0$

We see that $(0,0) ,(1,1) $ Satisfy equation.

The graph of the equation is drawn as shown.

The line divides the XY plane into two regions ( half plane)

Consider the point (1,0)

We see that $1-0 \leq 0$

$ 1 \leq 0$ is false

Thus the inequality (3) is represented by the region above the line $x-y=0$ not including the point (0,0) (not including the line )

Step 4:

The fourth inequality is $x \geq 0, y \geq 0$

It represents the first quadrant

Step 5:

Hence the solution of the system of linear inequalities is represented by the common shaded region in the first quadrant including the line.