Solving the system of equations graphically — lesson. Mathematics State Board, Class 9.

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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.

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4. Solving the system of equations graphically

Theory: