### Theory:

Let us consider the system of linear equations and find the solution graphically.

Find the solution to the system of equations $$2x+y=8$$ and $$4x+2y=10$$ graphically.

Solution:

Let us consider the equation $$2x+y=8$$. Now, we shall substitute the values for $$x$$ to find the value of $$y$$.

When $$x=1$$, $$2(1)+y=8$$$$\Rightarrow y=8-2$$$$\Rightarrow y=6$$

When $$x=2$$, $$2(2)+y=8$$$$\Rightarrow y=8-4$$$$\Rightarrow y=4$$

When $$x=3$$, $$2(3)+y=8$$$$\Rightarrow y=8-6$$$$\Rightarrow y=2$$

When $$x=4$$, $$2(4)+y=8$$$$\Rightarrow y=8-8$$$$\Rightarrow y=0$$

Now, writing these values in the table, we have:

 $$x$$ $$1$$ $$2$$ $$3$$ $$4$$ $$y$$ $$6$$ $$4$$ $$2$$ $$0$$

Similarly, plotting the points for the equation $$4x+2y=10$$, we have:

When $$x=0$$, $$4(0)+2y=10$$$$\Rightarrow 2y=10$$$$\Rightarrow y=5$$

When $$x=1$$, $$4(1)+2y=10$$$$\Rightarrow 2y=10-4=6$$$$\Rightarrow y=3$$

When $$x=2$$, $$4(2)+2y=10$$$$\Rightarrow 2y=10-8=2$$$$\Rightarrow y=1$$

When $$x=3$$, $$4(3)+2y=10$$$$\Rightarrow 2y=10-12=-2$$$$\Rightarrow y=-1$$

Now, writing these values in the table, we have:

 $$x$$ $$0$$ $$1$$ $$2$$ $$3$$ $$y$$ $$5$$ $$3$$ $$1$$ $$-1$$

Now, plotting these points in the graph, we have: Since the two lines in the graph do not intersect at any point, the graph is said to be an inconsistent system and has no solution.

Another method:

Consider writing the two equations $$2x+y=8$$ and $$4x+2y=10$$ in the form of $$y = mx + c$$ and determine the slope.

Solution:

The slope of the equation $$2x + y = 8$$ is given by:

$$y = -2x + 8$$

The slope of the equation $$2x + y = 8$$ is $$-2$$.

The slope of the equation $$4x + 2y = 10$$ is given by:

$$2y = -4x + 10$$

$$y = -2x + 5$$

The slope of the equation $$4x + 2y = 10$$ is $$-2$$.
Important!
When writing these two equations in the form of $$y = mx+c$$, both the equations have the same slope. This proves that the two lines are parallel and does not have a point of intersection.