Question

Solve the following system of equations graphically and find the vertices and area of the triangle formed by these lines and the y-axis.
$$\begin{gathered} 2x-5y+4=0 \\ 2x+y-8=0 \end{gathered}$$

Answer 1

From the first equation, write y in terms of x
$y=\frac{2x+4}{5}.....\left(i\right)$
Substitute different values of x in (i) to get different values of y
$$\begin{gathered} \mathrm{For}x=-2,y=\frac{-4+4}{5}=0 \\ \mathrm{For}x=0,y=\frac{0+4}{5}=\frac{4}{5} \\ \mathrm{For}x=3,y=\frac{6+4}{5}=2 \end{gathered}$$
Thus, the table for the first equation (2x − 5y + 4 = 0) is

x	−2	0	3
y	0	$\frac{4}{5}$	2

Now, plot the points A(−2,0), $B\left(0,\frac{4}{5}\right)$ and C(3,2) on a graph paper and join
A, B and C to get the graph of 2x − 5y + 4 = 0.
From the second equation, write y in terms of x
$y=8-2x.....\left(ii\right)$
Now, substitute different values of x in (ii) to get different values of y
$$\begin{gathered} \mathrm{For}x=0,y=8-0=8 \\ \mathrm{For}x=2,y=8-4=4 \\ \mathrm{For}x=4,y=8-8=0 \end{gathered}$$
So, the table for the second equation (2x + y − 8 = 0 ) is

x	0	2	4
y	8	4	0

Now, plot the points D(0,8), E(2,4) and F(4,0) on the same graph paper and join
D, E and F to get the graph of 2x + y − 8 = 0.



From the graph it is clear that, the given lines intersect at (3,2).
So, the solution of the given system of equations is (3,2).
The vertices of the triangle formed by the system of equations and y-axis are (0,8), $\left(0,\frac{4}{5}\right)$ and (3,2).
Draw a perpendicular from point C to the y-axis. So,
$$\begin{gathered} \mathrm{Area}\left(∆DBC\right)=\frac{1}{2}\times DB\times CM \\ \mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}=\frac{1}{2}\times \left(8-\frac{4}{5}\right)\times 3 \\ \mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}=\frac{54}{5}\mathrm{sq}.\mathrm{units} \end{gathered}$$
Hence, the veritices of the triangle are (0,8), $\left(0,\frac{4}{5}\right)$ and (3,2) and its area is $\frac{54}{5}$ sq. units.