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Home » Show graphically that the system of equations 3x – y = 5, 6x – 2y = 10 has infinitely many solutions.
Show graphically that the system of equations 3x – y = 5, 6x – 2y = 10 has infinitely many solutions.
CBSE | Class 10 | Excercise 3A | Maths | Pair of Linear Equations in Two Variables | RS Aggarwal
Show graphically that the system of equations 3x – y = 5, 6x – 2y = 10 has infinitely many solutions.

Solution:

Draw a horizontal line on a graph paper X’OX and a vertical line YOY’ representing the xaxis and y-axis, respectively.
Graph of 3 x-y=5
3 x-y=5
\Rightarrow y=3 x-5\dots \dots(i)

Putting x=1, we obtain y=-2
Putting x=0, we obtain y=-5
Putting x=2, we obtain y=1

Therefore, we have the following table for the eq. 3 x-y=5

    \[\begin{tabular}{|r|c|c|r|} \hline $\mathrm{x}$ & 1 & 0 & 2 \\ \hline $\mathrm{y}$ & $-2$ & $-5$ & 1 \\ \hline \end{tabular}\]

Now, plot the points \mathrm{A}(1,-2), \mathrm{B}(0,-5) and \mathrm{C}(2,1) on the graph paper.
Join \mathrm{AB} and \mathrm{AC} to get the graph line \mathrm{BC}. Extend it on both ways.

So, the line \mathrm{BC} is the graph of 3 \mathrm{x}-\mathrm{y}=5
Graph of 6 x-2 y=10
6 x-2 y=10
\begin{array}{l} \Rightarrow 2 \mathrm{y}=(6 \mathrm{x}-10) \\ \Rightarrow \mathrm{y}=\frac{6 x-10}{2}\dots \dots(ii) \end{array}

Putting x=0, we obtain y=-5
Putting x=1, we obtain y=-2
Putting x=2, we obtain y=1

So, we have the following table for the eq. 6 x-2 y=10.

    \[\begin{tabular}{|r|r|r|r|} \hline $\mathrm{x}$ & 0 & 1 & 2 \\ \hline $\mathrm{y}$ & $-5$ & $-2$ & 1 \\ \hline \end{tabular}\]

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 infinitely many solutions.

i

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