Solve the following systems of linear inequation graphically:

2*x* + 3*y* ≤ 6, *x* + 4*y* ≤ 4, *x* ≥ 0, *y* ≥ 0

#### Solution

Converting the inequations to equations, we obtain:

2*x* + 3*y* = 6, *x* + 4*y* = 4, *x* = 0, *y* = 0

2*x* + 3*y* =6: This line meets the *x-*axis at (3, 0) and the *y*-axis at (0, 2). Draw a thick line joining these points.

We see that the origin (0,0) satisfies the inequation 2*x* + 3*y* ≤ 6.

So, the portion containing the origin represents the solution set of the inequation 2*x* + 3*y* ≤ 6*x* + 4*y* = 4: This line meets the* x*-axis at (4, 0) and the *y-*axis at (0, 1). Draw a thick line joining these points.

We see that the origin (0,0) satisfies the inequation *x* + 4*y* ≤ 4.

So, the portion containing the origin represents the solution set of the inequation *x* + 4*y*≤ 4

Clearly, *x* ≥ 0, *y* ≥ 0 represents the first quadrant.

Hence, the shaded region in the figure represents the solution set of the given set of inequations.