By drawing a graph for each of the equations 3x + y + 5 = 0; 3y - x = 5 and 2x + 5y = 1 on the same graph paper; show that the lines given by these equations are concurrent (i.e. they pass through the same point).

Take 2 cm = 1 unit on both the axes.

First equation: 3x + y + 5 = 0

Step 1:

Give at least three suitable values to the variable x and find the corresponding values of y.

Let x = -3, then 3 $\times$ (-3) + y + 5 = 0 ⇒ y = 4

Let x = -2, then 3 $\times$ (-2) + y + 5 = 0 ⇒ y = 1

Let x = 1, then 3 $\times$ 1 + y + 5 = 0 ⇒ y = -8

Step 2:

Make a table (as given below) for the different pairs of the values of x and y:

Step 3:

Plot the points, from the table, on a graph paper and then draw a straight line passing through the points plotted on the graph.

Second equation: 3y - x = 5

Step 1:

Give at least three suitable values to the variable x and find the corresponding values of y.

Let x = -2, then 3y - (-2) = 5 ⇒ y = 1

Let x = 1, then 3y - 1 = 5 ⇒ y = 2

Let x = 7, then 3y - 7 = 5 ⇒ y = 4

Step 2:

Make a table (as given below) for the different pairs of the values of x and y:

Step 3:

Plot the points, from the table, on a graph paper and then draw a straight line passing through the points plotted on the graph.

Third equation: 2x + 5y = 1

Step 1:

Give at least three suitable values to the variable x and find the corresponding values of y.

Let x = -7, then 2 $\times$ (-7) + 5y = 1 ⇒ y = 3

Let x = -2, then 2 $\times$ (-2) + 5y = 1 ⇒ y = 1

Let x = 3, then 2 $\times$ 3 + 5y = 1 ⇒ y = -1

Step 2:

Make a table (as given below) for the different pairs of the values of x and y:

Step 3:

Plot the points, from the table, on a graph paper and then draw a straight line passing through the points plotted on the graph. 3

From the graph, it is clear that all three lines intersect at a common point (-2, 1), confirming that the lines are concurrent.