2x+3y>=6, x-y>=3 and y<=2 solve and graph

Question

solving systems of equations by graphing.

Answer 1

The inequalities are 2x + 3y  ≥ 6, x - y  ≥ 3 and y  ≤ 2.

Rewrite the inequalities so that solves for y  , That's the slope intercept form and it will make the boundary line easier to graph.

1) Draw the coordinate plane.

The first inequality 2x + 3y  ≥ 6 .

3y  ≥ - 2x + 6

y  ≥ (- 2/3)x + 2.

2) Graph the line y = (- 2/3)x + 2.

3) Since the inequality symbol is  ≥  the boundary is included in the solution set.

Graph the boundary of the inequality y  ≥ (- 2/3)x + 2  with solid line.

4) To determine which half plane to be shaded use a test point in either half- plane.

A simple chioce is (0,0). Substitute x  = 0 and y  = 0 in original inequality y  ≥ (- 2/3)x + 2.

0  ≥ (- 2/3)(0) + 2

0 ≥  2.

The statement is false.

5) Since the statement is false , shade the region does not contains the point (0, 0),Shaded in maroon colour.

The second inequality x - y ≥ 3.

- y ≥ - x + 3

y ≤ x - 3.

6) Graph the line y  = x - 3.

7) Since the inequality symbol is ≤ the boundary is included in the solution set.

Graph the boundary of the inequality y ≤ x - 3  with solid line.

8) To determine which half plane to be shaded use a test point in either half- plane.

A simple chioce is (0, 0). Substitute x  = 0 and y  = 0 in original inequality y ≤ x - 3.

0 ≤ 0 - 3

0 ≤ - 3.

The statement is false.

9) Since the statement is false , shade the region does not contains the  point (0, 0), shaded in blue colour.

The third inequality y ≤  2.

10) Graph the line y  = 2.

11) Since the inequality symbol is  ≤  the boundary is included in the solution set.

Graph the boundary of the inequality y ≤  2 with solid line.

12) To determine which half plane to be shaded use a test point in either half- plane.

A simple chioce is (0, 0). Substitute x  = 0 and y  = 0 in original inequality y ≤  2.

0 ≤  2

The statement is true.

13) Since the statement is true , shade the region contains the point (0,0), shaded in green colour.

Graph :

The solution of the system is the set of ordered pairs in the intersection of the graph of

2x + 3y  ≥ 6, x - y  ≥ 3 and y  ≤ 2. This region is shaded in dark purple color.