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?

Cevap: 12

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Linear Programming and Systems of Inequalities

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