Question

Given a system of inequation:

$2y-x\le 4$
$-2x+y\ge -4$

Find the value of $s$, which is the greatest possible sum of the $x$ and $y$ co-ordinates of the point which satisfies the following inequalities as graphed in the $xy$ plane.

• A. $8$
• B. $12$
• C. $2$
• D. $4$

Answer 1

The correct option is A $8$

First, rewrite each equation, so that it is in the slope-intercept form of a line, which is $y=mx+b$, where $m$ is the slope and $b$ is the $y$-intercept of the line.

The first equation becomes $2y<x+4$ or $y\le \frac{1}{2}x+2$.

The second equation becomes $y\ge 2x-4$.

The greatest $x+y$ is the point at which the two lines intersect.

Set the equations of the two lines, $y=\frac{1}{2}x+2$ and $y=2x-4$, equal to each other and solve for $x$.

The resulting equation is $y=\frac{1}{2}x+2$ and $y=2x-4$.

Solve for $x$ to get $y=\frac{3}{-2}x+2=-4$ or $\frac{3}{-2}x=-6$,

$\Rightarrow x=4$

Next, plug $4$ into one of the two equations to solve for $y$.

Therefore, $y=2\left(4\right)-4=4$ and $x+y=4+4=8$.

The correct answer is $8$.