Answer
The coordinates of the line of the equation $2x+y=8$ are $\left( 1,6 \right),\left( 3,2 \right)$
Work Step by Step
Let us assume that x is the length of the rectangle and y is the breadth of the given rectangle.
So, the first condition is the equation of the line written as:
$2x+y=8$ (I)
And the second condition is that the area of rectangle, which is $6$ square units, can be represented as the product of its length and its width as given below:
$ x\cdot y=6$ (II)
Therefore, from equation (II), the value of y can be determined:
$ y=\frac{6}{x}$
Now, put $ y=\frac{6}{x}$ in equation (I) and solve for the value of x:
$\begin{align}
& 2x+y=8 \\
& 2x+\frac{6}{x}=8 \\
& 2{{x}^{2}}+6=8x \\
& 2{{x}^{2}}-8x+6=0
\end{align}$
Taking $2$ as a common factor from the above equation:
$\begin{align}
& 2\left( {{x}^{2}}-4x+3 \right)=0 \\
& {{x}^{2}}-3x-x+3=0 \\
& x\left( x-3 \right)-1\left( x-3 \right)=0 \\
& \left( x-1 \right)\left( x-3 \right)=0
\end{align}$
This implies that $ x=1$ and $ x=3$.
Now, substitute the values of x into $ y=\frac{6}{x}$.
When $ x=1$, $ y=6$.
And when $ x=3$, $ y=2$
Thus, the coordinates of the line of equation $2x+y=8$ are $\left( 1,6 \right)$ and $\left( 3,2 \right)$.
