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Find the coordinates of the vertices of the figure formed by the system of inequalities.x ≥ 0y ≤ 0–3x + y = –6

A. (0, 0), (1, 0), (–3, 0)

B. (0, 0), (2, 0), (0, –6)

C. (0, 0), (–3, 0), (0, –6)

D. (0, 0), (2, 0), (–3, 0)

5-Minute Check 3Optimization with Linear Programming

A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.

Mathematical Practices

4 Model with mathematics.

8 Look for and express regularity in repeated reasoning.

CCSSYou solved systems of linear inequalities by graphing.

- Find the maximum and minimum values of a function over a region.

- Solve real-world optimization problems using linear programming.

Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function f(x, y) = 3x – 2y for this region.

x≤ 5

y ≤ 4

x + y ≥ 2

Bounded Region

Step 1Graph the inequalities.

The polygon formed is atriangle with vertices at (–2, 4), (5, –3), and (5, 4).

Example 1Step 2Use a table to find the maximum and minimum values of f(x, y). Substitute the coordinates of the vertices into the function.

← minimum

← maximum

Answer: The vertices of the feasible region are (–2, 4), (5, –3), and (5, 4). The maximum value is 21 at (5, –3). The minimum value is –14 at (–2, 4).

Example 1Graph the following system of inequalities. What are the maximum and minimum values of the function f(x, y) = 4x – 3y for the feasible region of the graph?x≤ 4 y ≤ 5 x + y ≥ 6

A. maximum: f(4, 5) = 5minimum: f(1, 5) = –11

B. maximum: f(4, 2) = 10minimum: f(1, 5) = –11

C. maximum: f(4, 2) = 10minimum: f(4, 5) = 5

D. maximum: f(1, 5) = –11minimum: f(4, 2) = 10

Example 1 Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function f(x, y) = 2x + 3y for this region.

–x + 2y ≤ 2

x – 2y ≤ 4

x + y ≥ –2

Unbounded Region

Graph the system of inequalities. There are only two points of intersection, (–2, 0) and (0, –2).

Example 2 The minimum value is –6 at (0, –2). Although f(–2, 0) is –4, it is not the maximum value since there are other points that produce greater values. For example, f(2,1) is 7 and f(3, 1) is 9. It appears that because the region is unbounded, f(x, y) has no maximum value.

Unbounded Region

Answer: The vertices are at (–2, 0) and (0, –2). There is no maximum value. The minimum value is –6 at (0, –2).

Example 2Graph the following system of inequalities. What are the maximum and minimum values of the function f(x, y) = x + 2y for the feasible region of the graph?x + 3y ≤ 6 –x – 3y ≤ 9 2y – x ≥ –6

A. maximum: no maximumminimum: f(6, 0) = 6

B. maximum: f(6, 0) = 6minimum: f(0, –3) = –6

C. maximum: f(6, 0) = 6minimum: no minimum

D. maximum: no maximumminimum: f(0, –3) = –6

Example 2Pg 157: 2-12 even

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