Question
Maximize $Z = 9x + 3y$
Subject to
$2\text{x}+3\text{y}\leq13$
$3\text{x}+\text{y}\leq5$
$\text{x},\text{y}\geq0$
Subject to
$2\text{x}+3\text{y}\leq13$
$3\text{x}+\text{y}\leq5$
$\text{x},\text{y}\geq0$
✓
Answer
Coverting the given inequation into equations, we get
$2x + 3y = 13, 3x + y = 5$ and $x = 0, y = 0$
Region represented by $2\text{x}+3\text{y}\leq13:$
The line meets coordinate axes at $\text{A}_1\Big(\frac{13}{2},0\Big)$ and $\text{B}_1\Big(0,\frac{13}{3}\Big)$ respectively.
Join these points to obtain the line $2x + 3y = 13,$ clearly, $(0,0)$ satisfies the in eqation $2\text{x}+3\text{y}\leq13$,
so, the region in xy-plane that contains origin represents the solution set of $2\text{x}+3\text{y}\leq13$.
Region represented by $3\text{x}+\text{y}\leq5:$
The line meets coordinate axes at $\text{A}_2\Big(\frac{5}{3},0\Big)$ and $B_2(0, 5)$ respectively.
Join these points to obtain the line $3x + y = 5$, clearly, $(0, 0)$ satisfies the in eqation $3\text{x}+\text{y}\leq5$,
so, the region in xy-plane that contains origin represents the solution set of $3\text{x}+\text{y}\leq5$.
Region represented by $\text{x},\text{y}\geq0:$
It clearly represent first quadrant of xy-plane.
The common region to regions represented by above in equalities.
The coordinates of the corner points of the shaded region are $
\text{O}(0,0),\text{A}\Big(\frac{5}{3},0\Big),\text{P}\Big(\frac{2}{7},\frac{29}{7}\Big),\text{B}_2\Big(0,\frac{13}{3}\Big)$.
The value of $Z = 9x + 3y$ at
$\text{O}(0,0)=9(0)+3(0)=0$
$\text{A}_1\Big(\frac{5}{3},0\Big)=9\Big(\frac{5}{3}\Big)+3(0)=15$
$\text{P}\Big(\frac{2}{7},\frac{29}{7}\Big)=9\Big(\frac{2}{7}\Big)+3\Big(\frac{29}{7}\Big)=15$
$\text{B}_2\Big(0,\frac{13}{3}\Big)=9(0)+3\Big(\frac{13}{3}\Big)=13$
Clearly, Z is maximum at every point on the line joining $A_1$ and P, So
$\text{x}=\frac{2}{7}$ or $\frac{2}{7}$, $\text{y}=0$ or $\frac{29}{7}$
and maximum $Z = 15$.
$2x + 3y = 13, 3x + y = 5$ and $x = 0, y = 0$
Region represented by $2\text{x}+3\text{y}\leq13:$
The line meets coordinate axes at $\text{A}_1\Big(\frac{13}{2},0\Big)$ and $\text{B}_1\Big(0,\frac{13}{3}\Big)$ respectively.
Join these points to obtain the line $2x + 3y = 13,$ clearly, $(0,0)$ satisfies the in eqation $2\text{x}+3\text{y}\leq13$,
so, the region in xy-plane that contains origin represents the solution set of $2\text{x}+3\text{y}\leq13$.
Region represented by $3\text{x}+\text{y}\leq5:$
The line meets coordinate axes at $\text{A}_2\Big(\frac{5}{3},0\Big)$ and $B_2(0, 5)$ respectively.
Join these points to obtain the line $3x + y = 5$, clearly, $(0, 0)$ satisfies the in eqation $3\text{x}+\text{y}\leq5$,
so, the region in xy-plane that contains origin represents the solution set of $3\text{x}+\text{y}\leq5$.
Region represented by $\text{x},\text{y}\geq0:$
It clearly represent first quadrant of xy-plane.
The common region to regions represented by above in equalities.
The coordinates of the corner points of the shaded region are $
\text{O}(0,0),\text{A}\Big(\frac{5}{3},0\Big),\text{P}\Big(\frac{2}{7},\frac{29}{7}\Big),\text{B}_2\Big(0,\frac{13}{3}\Big)$.
The value of $Z = 9x + 3y$ at
$\text{O}(0,0)=9(0)+3(0)=0$
$\text{A}_1\Big(\frac{5}{3},0\Big)=9\Big(\frac{5}{3}\Big)+3(0)=15$
$\text{P}\Big(\frac{2}{7},\frac{29}{7}\Big)=9\Big(\frac{2}{7}\Big)+3\Big(\frac{29}{7}\Big)=15$
$\text{B}_2\Big(0,\frac{13}{3}\Big)=9(0)+3\Big(\frac{13}{3}\Big)=13$
Clearly, Z is maximum at every point on the line joining $A_1$ and P, So
$\text{x}=\frac{2}{7}$ or $\frac{2}{7}$, $\text{y}=0$ or $\frac{29}{7}$
and maximum $Z = 15$.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Solve graphically :x + 2y ≤ 6
The rate of increase of bacteria in a culture is proportional to the number of bacteria present and it is found that the number doubles in 6 hours. Prove that the bacteria becomes 8 times at the end of 18 hours.
If $\text{y}=\text{e}^{\tan^{-1}\text{x}},$ prove that $(1+\text{x}^2)\text{y}_2+(2\text{x}-1)\text{y}_1=0$
→Evaluvate the following intregals:
$\int\frac{1}{3+4\cot\text{x}}\ \text{dx}$
→$\int\frac{1}{3+4\cot\text{x}}\ \text{dx}$
If $\text{x}=\text{a}\sec\theta,\text{y}=b\tan\theta$ prove that $\frac{\text{d}^2\text{y}}{\text{dx}^2}=-\frac{\text{b}^4}{\text{a}^2\text{y}^3}$
→A square piece of tin of side $18\ cm$ is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut off so that the volume of the box is maximum? Find this maximum volume.
→Evaluate the following:
$\sin\Big(\sec^{-1}\frac{17}{8}\Big)$
→$\sin\Big(\sec^{-1}\frac{17}{8}\Big)$
Solve the following differential equations:
→$\frac{d y}{d x}=\frac{2 y-x}{2 y+x}$
If $\text{y}=\sqrt{\text{a}^2-\text{x}^2},$ prove that $\text{y}\frac{\text{dy}}{\text{dx}}+\text{x}=0$
→Find the intervals in which the following functions are increasing or decreasing.
$\text{f}(\text{x})=\frac{3}{2}\text{x}^4-4\text{x}^3-45\text{x}^2+51$
→$\text{f}(\text{x})=\frac{3}{2}\text{x}^4-4\text{x}^3-45\text{x}^2+51$