In the xy-plane, a point with coordinates (x, y) lies in the solution set of the system of… — SAT

In the

xy

-plane, a point with coordinates

(x, y)

lies in the solution set of the system of inequalities below:

\begin{aligned} 2(x - 5) + y &\leq 16 \\ x + y &\leq 25 \end{aligned}

y \geq 8

, what is the maximum possible integer value of

x

To find the maximum possible integer value of

x

in the system's solution set under the constraint

y \geq 8

, we find the boundaries for

x

in both inequalities. Simplifying the first inequality,

2(x - 5) + y \leq 16

, gives

2x - 10 + y \leq 16

, which simplifies to

x \leq 13 - 0.5y

. The second inequality,

x + y \leq 25

, simplifies to

x \leq 25 - y

. Since both boundary lines have negative slopes, the maximum value of

x

for any point in the solution set occurs at the minimum value of

y

, which is

y = 8

. Substituting

y = 8

into both simplified inequalities gives

x \leq 9

and

x \leq 17

. To satisfy both inequalities simultaneously,

x

must be less than or equal to the more restrictive upper bound, which is 9. Thus, the maximum possible integer value of

x

is 9.

Simplify the first inequality to isolate x.

2(x - 5) + y \leq 16 \implies 2x - 10 + y \leq 16 \implies 2x \leq 26 - y \implies x \leq 13 - 0.5y

To determine the boundary condition imposed on x by the first inequality.

Simplify the second inequality to isolate x.

x + y \leq 25 \implies x \leq 25 - y

To determine the boundary condition imposed on x by the second inequality.

Determine the maximum value of x under the condition

y \geq 8

For the first inequality, the maximum value of x occurs at the minimum value of y (

y = 8

), which gives

x \leq 13 - 0.5(8) = 9

. For the second inequality, the maximum value of x occurs at

y = 8

, which gives

x \leq 25 - 8 = 17

Because both boundary lines have negative slopes, the maximum value of x is achieved when y is at its minimum.

Find the intersection of the constraints to find the maximum possible value of x that satisfies both inequalities.

Since x must be less than or equal to 9 and less than or equal to 17, the maximum integer value that satisfies both conditions is 9.

The solution set must satisfy all inequalities in the system simultaneously.

Systems of Linear Inequalities in Two Variables

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