Show graphically that the system of equations 2x + y = 6, 6x + 3y = 18 has infinitely many solutions.
Show graphically that the system of equations 2x + y = 6, 6x + 3y = 18 has infinitely many solutions.

Solution:

Draw a horizontal line on a graph paper \mathrm{X}^{\prime} \mathrm{OX} and a vertical line YOY’ representing the \mathrm{x}- axis and y-axis, respectively.
Graph of 2 \mathbf{x}+\mathbf{y}=6
2 \mathrm{x}+\mathrm{y}=6
\Rightarrow \mathrm{y}=(6-2 \mathrm{x}) \quad \ldots(\mathrm{i})

Putting x=3, we obtain y=0
Putting x=1, we obtain y=4
Putting x=2, we obtain y=2

Therefore, we have the following table for the eq. 2 \mathrm{x}+\mathrm{y}=6

    \[\begin{tabular}{|r|r|r|r|} \hline $\mathrm{x}$ & 3 & 1 & 2 \\ \hline $\mathrm{y}$ & 0 & 4 & 2 \\ \hline \end{tabular}\]

Now, plot the points \mathrm{A}(3,0), \mathrm{B}(1,4) and \mathrm{C}(2,2) on the graph paper. Join \mathrm{AC} and \mathrm{CB} to get the graph line \mathrm{AB}. Extend it on both ways.

Therefore, the line AB is the graph of 2 \mathrm{x}+\mathrm{y}=6
Graph of 6 x+3 y=18
\begin{array}{l} 6 x+3 y=18 \\ \Rightarrow 3 y=(18-6 x) \\ \Rightarrow y=\frac{18-6 x}{3}\dots \dots (ii) \end{array}

Putting x=3, we obtain y=0
Putting x=1, we obtain y=4
Putting x=2, we obtain y=2

So, we have the following table for the equation 6 \mathrm{x}+3 \mathrm{y}=18.

    \[\begin{tabular}{|r|r|r|r|} \hline $\mathrm{x}$ & 3 & 1 & 2 \\ \hline $\mathrm{y}$ & 0 & 4 & 2 \\ \hline \end{tabular}\]

These are the same points as obtained for the graph line of eq.(i).


It is clear from the graph that these two lines coincide.
As a result, the given system of equations has an infinite number of solutions.