Question

If $S$ is the solution region of the inequalities $\{\begin{array}{c}x+3y\le 18\\ 2x-3y\le 9\end{array}$ and $(a,b)\in S$, then the minimum possible value for $b$ is

Answer 1

If $(a,b)$ is a solution to the system, then $a$ is the $x$-coordinate of any point in the darkest shaded region and $b$ is the corresponding $y$-coordinate.

When $a=6$, the minimum possible value for $b$ lies on the lower boundary line, $2x-3y\le 9$.

It looks like the $y$-coordinate is $1$, but to be sure, substitute $x=6$ into the equation and solve for $y$.

You can use $=$ in the equation, instead of the inequality symbol, because you are finding a point on the boundary line.

$2x–3y=9$

$\Rightarrow 2\left(6\right)–3y=9$

$\Rightarrow 12–3y=9$

$\Rightarrow -3y=-3$

$\Rightarrow y=1$

So, the value of $b$ is $1$.