Q.Solve the following linear programming problem graphically: Maximize $Z = 4x + 3y$ subject to the constraints $x + 2y \le 10,\ 3x + y \le 15,\ x, y \ge 0$.

Constraints: $x+2y\le10,\ 3x+y\le15,\ x,y\ge0$.

Corner points. At $y=0$: line $x+2y=10$ gives $x\le10$; line $3x+y=15$ gives $x\le5$; the binding bound is $x\le5$, giving vertex $(5,0)$ — check the first line: $5\le10$, slack.

At $x=0$: first line gives $y\le5$; second gives $y\le15$; binding is $y\le5$, giving vertex $(0,5)$ — check second line: $5\le15$, slack.

Intersection: $x+2y=10$ and $3x+y=15$. From the first, $y=\dfrac{10-x}{2}$; substituting, $3x+\dfrac{10-x}{2}=15\Rightarrow6x+10-x=30\Rightarrow5x=20\Rightarrow x=4,\ y=\dfrac{10-4}{2}=3$, giving $(4,3)$. …

Find the x- and y-intercepts of each boundary line, determine which is binding at each axis by comparing the two, solve the two boundary equations simultaneously for the interior corner, t …

Using the intercept of only one line at each axis without checking whether the other constraint is more restrictive there; arithmetic slips when …