Question

Solve the following system of inequalities graphically:
$4x+3y\le 60,y\ge 2x,x\ge 3,x\ge 0,y\ge 0$

Answer 1

Solving inequations for graph
Given inequations
$4x+3y\le 60,y\ge 2x,x\ge 3,x\ge 0,y\ge 0$
First, we solve $4x+3y\le 60....\left(1\right)$

$\begin{array}{cc}\text{Putting}x=0\text{in}& \text{Putting}y=0\text{in}\\ \text{equation (1)}& \text{equation (1)}\\ \Rightarrow 4\left(0\right)+3y=60& \Rightarrow 4x+3\left(0\right)=60\\ \Rightarrow y=\frac{60}{3}& \Rightarrow x=\frac{60}{4}\\ \Rightarrow y=20& \Rightarrow x=15\end{array}$

Hence,
$\begin{array}{ccc}x& 0& 15\\ y& 20& 0\end{array}$

Now, we solve $y\ge 2x$
Let's draw graph $y=2x....\left(2\right)$

$\begin{array}{cc}\text{Putting}x=0\text{in}& \text{Putting}x=2\text{in}\\ \text{equation (2)}& \text{equation (2)}\\ \Rightarrow y=2\left(0\right)& \Rightarrow y=2\left(2\right)\\ \Rightarrow y=0& \Rightarrow y=4\end{array}$

Hence,
$\begin{array}{ccc}x& 0& 2\\ y& 0& 4\end{array}$

Also, $x\ge 3$
Let's draw graph of $x=3$
So, for all values of $y,x=3$
$\begin{array}{cccc}x& 3& 3& 3\\ y& -2& 0& 3\end{array}$

Drawing graph,

Solve for required region:
$4x+3y\le 60,y\ge 2x,x\ge 3,x\ge 0,y\ge 0$

Also, $x\ge 0\&y\ge 0$
So, our shaded region will lie in first quadrant.

Taking common region that satisfies all the given inequalities.