A bounded region in the xy-plane is defined by the following system of inequalities: \begi… — SAT

A bounded region in the $xy$ -plane is defined by the following system of inequalities:

\begin{aligned} y &\ge \frac{1}{2}x + 2 \\ y &\le 2x - 1 \\ y &\le -x + 11 \end{aligned}

What is the maximum $y$ -coordinate of any point within this region?

Answer: 7

Answer

The maximum y-coordinate of any point in the bounded region is 7.

The solution region is a triangle with vertices at

(2, 3)

(4, 7)

, and

(6, 5)

. Since all inequalities are less-than-or-equal-to or greater-than-or-equal-to, the boundary points are included in the solution set. The maximum y-value occurs at the vertex

(4, 7)

, which gives a maximum y-coordinate of 7.

Step-by-Step Solution

Determine the equations of the boundary lines.

The boundary equations are

y = \frac{1}{2}x + 2

y = 2x - 1

, and

y = -x + 11

These equations represent the boundaries of the system of inequalities.

Find the intersection point of

y = 2x - 1

and

y = \frac{1}{2}x + 2

Solving

2x - 1 = \frac{1}{2}x + 2

yields

x = 2

, which gives

y = 3

. The intersection is

(2, 3)

This is one vertex of the bounded region.

Find the intersection point of

y = 2x - 1

and

y = -x + 11

Solving

2x - 1 = -x + 11

yields

x = 4

, which gives

y = 7

. The intersection is

(4, 7)

This is the second vertex of the bounded region.

Find the intersection point of

y = -x + 11

and

y = \frac{1}{2}x + 2

Solving

-x + 11 = \frac{1}{2}x + 2

yields

x = 6

, which gives

y = 5

. The intersection is

(6, 5)

This is the third vertex of the bounded region.

Compare the y-coordinates of the three vertices of the bounded region.

The y-coordinates of the vertices

(2, 3)

(4, 7)

, and

(6, 5)

are 3, 7, and 5, respectively. The maximum value is 7.

The maximum value of a linear coordinate over a bounded convex region occurs at one of its vertices.

Key Concept

Finding the vertices of a bounded region defined by a system of linear inequalities and optimizing a coordinate value.

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