A closed triangular region in the coordinate plane is defined by the following system of l… — SAT

A closed triangular region in the coordinate plane is defined by the following system of linear inequalities:

\begin{aligned} y &\geq 2x - 4 \\ y &\leq -x + 8 \\ x &\geq 1 \end{aligned}

What is the maximum possible value of the expression $2x + y$ for any point $(x, y)$ that lies within or on the boundary of this region?

Answer: 12

Answer

The maximum possible value of the expression

2x + y

is 12.

To find the maximum possible value of the expression

2x + y

subject to the given system of inequalities, we first identify the boundary lines and find the vertices of the bounded triangular region in the coordinate plane. The boundary lines are

y = 2x - 4

y = -x + 8

, and

x = 1

. The intersection of

y = 2x - 4

and

y = -x + 8

occurs at

x = 4, y = 4

, which gives vertex

(4, 4)

. The intersection of

y = 2x - 4

and

x = 1

occurs at

(1, -2)

. The intersection of

y = -x + 8

and

x = 1

occurs at

(1, 7)

. Evaluating the linear expression

2x + y

at these three vertices gives

2(4) + 4 = 12

2(1) - 2 = 0

, and

2(1) + 7 = 9

. By the corner point theorem, the maximum value of a linear function on a closed polygonal region occurs at one of the vertices. Comparing the values, the maximum possible value is 12.

Step-by-Step Solution

Find the vertex formed by the intersection of the boundary lines

y = 2x - 4

and

y = -x + 8

Vertex

A(4, 4)

Setting the two equations equal:

2x - 4 = -x + 8 \implies 3x = 12 \implies x = 4

. Substituting

x = 4

back into either equation yields

y = 4

Find the vertex formed by the intersection of the boundary line

y = 2x - 4

and the vertical line

x = 1

Vertex

B(1, -2)

Substituting

x = 1

into

y = 2x - 4

gives

y = 2(1) - 4 = -2

Find the vertex formed by the intersection of the boundary line

y = -x + 8

and the vertical line

x = 1

Vertex

C(1, 7)

Substituting

x = 1

into

y = -x + 8

gives

y = -1 + 8 = 7

Evaluate the expression

2x + y

at each of the three vertices.

A(4, 4)

, the value is

12

; at

B(1, -2)

, the value is

0

; at

C(1, 7)

, the value is

9

According to the Corner Point Theorem of linear programming, the maximum or minimum of a linear objective function on a closed bounded region must occur at one of the vertices.

Identify the maximum value from the evaluated points.

The maximum value is 12.

Comparing the values 12, 0, and 9 shows that 12 is the largest value.

Key Concept

Linear Programming and Systems of Inequalities

Question

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