A system of inequalities is shown below. \begin{aligned} y &\leq -3x + 15 \\ y &\leq 2x + 2 \end{aligned} If (x, y) is a solution to the system, what is the maximum possible integer value of y?

A system of inequalities is shown below. \begin{aligned} y &\leq -3x + 15 \\ y &\leq 2x + … — SAT

A system of inequalities is shown below.

\begin{aligned} y &\leq -3x + 15 \\ y &\leq 2x + 2 \end{aligned}

(x, y)

is a solution to the system, what is the maximum possible integer value of

y

A
6
7Answer
C
8
D
15

Answer

The maximum value of

y

in the system occurs at the intersection of the boundary lines

y = -3x + 15

and

y = 2x + 2

. Solving for the intersection gives

x = 2.6

and

y = 7.2

. Since any point in the solution set must satisfy

y \leq 7.2

, the largest possible integer value for

y

is 7. We can confirm this is achievable because when

y = 7

x

can be any value in the range

[2.5, 2.67]

(such as

x=2.6

), which satisfies both inequalities.

Step-by-Step Solution

Find the intersection point of the boundary lines.

Set the two equations equal to find the

x

-coordinate:

-3x + 15 = 2x + 2 \implies 5x = 13 \implies x = 2.6

The maximum value of

y

in a system bounded from above by two lines with opposing slopes occurs at their point of intersection.

Calculate the corresponding maximum

y

-value.

Substitute

x = 2.6

into either boundary equation:

y = 2(2.6) + 2 = 7.2

This determines the absolute upper bound of

y

for any coordinate pair in the solution set.

Determine the maximum integer value of

y

Since

y \leq 7.2

, the largest integer value that

y

can take is 7.

The question asks specifically for the maximum integer value of

y

, and since

7 \leq 7.2

, a solution exists at this

y

-value.

Verify that a solution exists for

y = 7

For

y = 7

, the system requires

7 \leq -3x + 15 \implies x \leq 2.67

and

7 \leq 2x + 2 \implies x \geq 2.5

. The interval

[2.5, 2.67]

is non-empty, so valid solutions (e.g.,

(2.6, 7)

) exist.

This confirms that

y = 7

is attainable within the feasible region.

Key Concept

Optimization of variables within systems of linear inequalities

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