Question

Solve the following system of linear equations graphically
$3x+y-11=0$
$x-y-1=0$
Shade the region bounded by these lines and $y-$axis. Also find the area of this region.

Answer 1

Let the first line

$3x+y-11=0$

$y=11-3x$

x	0	1	2	$\frac{11}{3}$
y	11	8	5	0

so the points $(0,11),(1,8),(2,5),(\frac{11}{3},0)$

and

$x-y-1=0$

$y=1-x$

x	0	1	2
y	1	0	-1

so points are

$(0,1),(1,0),(2,-1)$

Let the quadrilateral $ABCD$ is required area and points are $A(0,11),B(0,1),C(1,0),D(\frac{11}{3},0)$ an the origin at $E(0,0)$

Area of quad. $ABCD=$ area of $\Delta AED-$ area of $\Delta BEC$ ----- (i)

Area of triangle $AED=\frac{Ax(E_y-D_y)+E_x(D_y-A_y)+D_x(A_y-E_y)}{2}$

$=\frac{0(0-0)+(0-11)+\frac{11}{3}(11-0)}{2}=\frac{121}{6}$ sq.units

similarly,

Area of $\Delta BEC=\frac{0+0+1(1-0)}{2}=\frac{1}{2}$

Now from (i)

$=\frac{121}{6}-\frac{1}{2}$

$=\frac{118}{6}$

Hence the required area

$\frac{118}{6}$ sq.units