In the xy-plane, a point with coordinates (x, y) lies in the region defined by the system … — SAT

In the $xy$ -plane, a point with coordinates $(x, y)$ lies in the region defined by the system of inequalities below.

y + 2x \leq 12

x - 2y \leq 6

x \geq 2

What is the maximum possible value of $y$ ?

Cevap: 8

Cevap

The system of inequalities bounds the solution set. The upper boundary is given by

y \leq -2x + 12

. Because the slope is negative, the maximum value of

y

on this boundary occurs at the smallest possible value of

x

. The constraint

x \geq 2

dictates that the minimum value of

x

is 2. Substituting

x = 2

into the boundary equation yields

y = -2(2) + 12 = 8

. Checking this coordinate against the third inequality,

2 - 2(8) = -14 \leq 6

, verifies that

(2, 8)

is a valid solution.

Adım Adım Çözüm

Express the first two inequalities in terms of y.

y \leq -2x + 12

and

y \geq \frac{1}{2}x - 3

This helps identify the upper and lower boundaries of the solution region.

Determine the boundary line that limits the maximum values of y.

The upper boundary is the line

y = -2x + 12

Since the inequality is

y \leq -2x + 12

, any solution must lie on or below this line.

Find the maximum value of y on this boundary line given the constraint

x \geq 2

y \leq -2(2) + 12 = 8

Since the slope of the boundary line is negative, y is maximized when x is at its minimum value, which is 2.

Verify that the point

(2, 8)

satisfies the inequality

x - 2y \leq 6

2 - 2(8) = -14 \leq 6

, which is true.

This confirms that the point

(2, 8)

is indeed in the solution set of the system.

Anahtar Kavram

Maximizing a coordinate value subject to a system of linear inequalities in two variables

Alternatif Yöntem

Instead of graphing or checking boundaries, we can algebraically solve for the boundary. Since

x \geq 2

, multiplying by

-2

and reversing the inequality gives

-2x \leq -4

. Adding 12 to both sides gives

-2x + 12 \leq 8

. Since

y \leq -2x + 12

, we get

y \leq 8

. Checking if

y = 8

and

x = 2

satisfies the second inequality

x - 2y \leq 6

confirms that

2 - 16 = -14 \leq 6

is true, meaning

y = 8

is indeed a valid solution and thus the maximum.

Tahmini Süre:1m 30s

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