Graphical Representation of a Pair of Linear Equations with Infinitely Many Solutions

Video Explanation

Question

Show graphically that the following system of equations has infinitely many solutions:

\[ 2x + 3y = 6 \]

\[ 4x + 6y = 12 \]

Solution

Step 1: Write Both Equations

Equation (1):

\[ 2x + 3y = 6 \]

Equation (2):

\[ 4x + 6y = 12 \]

Step 2: Compare the Two Equations

Dividing Equation (2) by 2:

\[ \frac{4x + 6y = 12}{2} \Rightarrow 2x + 3y = 6 \]

Thus, both equations represent the same line.

Step 3: Prepare Table of Values

For the Equation \(2x + 3y = 6\)

x	y
0	2
3	0

(The same table applies to both equations.)

Step 4: Graphical Representation

Plot the points \((0, 2)\) and \((3, 0)\) on the Cartesian plane.

Join these points to obtain a straight line.

Since both equations represent the same line, their graphs coincide completely.

Conclusion

As the two lines coincide, the given system of equations has infinitely many solutions.

Every point on the common line satisfies both equations.