Solve the following systems of linear equations graphically:
$2 x + 3 y = 6 4 x + 6 y = 12$

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answer is D.

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Detailed Solution

Drawing a graph for each equation is the first step in using the graphical technique to solve a system of equations or a set of simultaneous equations. Next, we search for points where the two graphs overlap. The system of equations will have a solution that is the coordinates of the point of intersection. The answer is such that it fulfils both requirements. The system of equations has no solution if the two lines become parallel to one another, i.e., they do not intersect, and an infinite number of solutions if the lines coincide. The set of equations presented here is as follows:

2 x + 3 y = 6

……….(`1)

4 x + 6 y = 12

……….(2)
For the first line

i . e . 2 x + 3 y = 6

we can give values of y corresponding to different values of x as follows:
When we take x =3, we can write from equation (1):

2 \times 3 + 3 y = 6

\Rightarrow 6 + 3 y = 6

\Rightarrow 3 y = 6 - 6

\Rightarrow y = 0

When we take x= -3, we have:

2 \times (- 3) + 3 y = 6

\Rightarrow - 6 + 3 y = 6

\Rightarrow 3 y = 12

\Rightarrow y = 4

So, two points lying on the 1st line are (3, 0) and (-3, 4).
Similarly for the second line

i . e . 4 x + 6 y = 12

we can give values of y corresponding to different values of x as follows:
When we take x= 6, we can write from equation (2):

4 \times 6 + 6 y = 12

\Rightarrow 24 + 6 y = 12

\Rightarrow 6 y = - 12

\Rightarrow y = - 2

When we take x= -3, we have:

4 \times (- 3) + 6 y = 12

\Rightarrow - 12 + 6 y = 12

\Rightarrow 6 y = 24

\Rightarrow y = 4

So, two points lying on the 2nd line are (6, -2) and (-3, 4).
So, the graph of the given lines can be made as:
Question Image

We can see that the graph of both the lines coincide with each other.
Hence, there are infinitely many solutions of the given system of equations.
Correct option is 4.

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