Solve, graphically, the following pairs of equations :

(i)

x - 5 = 0
y + 4 = 0

(ii)

2x + y = 23
4x - y = 19

(iii)

3x + 7y = 27

$8 - y = \dfrac{5}{2}x$

(iv)

$\dfrac{x + 1}{4} = \dfrac{2}{3}(1 - 2y)$

$\dfrac{2 + 5y}{3} = \dfrac{x}{7} - 2$

(i) x - 5 = 0

⇒ x = 5

and, y + 4 = 0

⇒ y = - 4

On the same graph paper, draw the graph for each given equation.

Both the straight lines drawn meet at point P. As it is clear from the graph, co-ordinates of the common point are (5, -4).

Solution of the given equations is : x = 5 and y = -4.

(ii)

First equation : 2x + y = 23

Step 1:

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

Let x = 2, then 2 $\times$ 2 + y = 23 ⇒ y = 19

Let x = 4, then 2 $\times$ 4 + y = 23 ⇒ y = 15

Let x = 6, then 2 $\times$ 6 + y = 23 ⇒ y = 11

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 : 4x - y = 19

Step 1:

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

Let x = 4, then 4 $\times$ 4 - y = 19 ⇒ y = -3

Let x = 6, then 4 $\times$ 6 - y = 19 ⇒ y = 5

Let x = 8, then 4 $\times$ 8 - y = 19 ⇒ y = 13

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.

On the same graph paper, draw the graph for each given equation.

Both the straight lines drawn meet at point P. As it is clear from the graph, co-ordinates of the common point are (7, 9).

Solution of the given equations is : x = 7 and y = 9.

(iii)

First equation : 3x + 7y = 27

Step 1:

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

Let x = 0, then 3 $\times$ 0 + 7y = 27 ⇒ y = 3.8

Let x = 2, then 3 $\times$ 2 + 7y = 27 ⇒ y = 3

Let x = 4, then 3 $\times$ 4 + 7y = 27 ⇒ y = 2.1

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.

Second equation : $8 - y = \dfrac{5}{2}x$

Step 1:

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

Let x = 0, then $8 - y = \dfrac{5}{2} \times 0$ ⇒ y = 8

Let x = 1, then $8 - y = \dfrac{5}{2} \times 1$ ⇒ y = 5.5

Let x = 2, then $8 - y = \dfrac{5}{2} \times 2$ ⇒ y = 3

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.

On the same graph paper, draw the graph for each given equation.

Both the straight lines drawn meet at point P. As it is clear from the graph, co-ordinates of the common point are (2, 3).

Solution of the given equations is : x = 2 and y = 3.

(iv)

First equation : $\dfrac{x + 1}{4} = \dfrac{2}{3}(1 - 2y)$

Step 1:

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

Let x = 0, then $\dfrac{0 + 1}{4} = \dfrac{2}{3}(1 - 2y)$ ⇒ y = 0.3

Let x = 3, then $\dfrac{3 + 1}{4} = \dfrac{2}{3}(1 - 2y)$ ⇒ y = -0.2

Let x = 7, then $\dfrac{7 + 1}{4} = \dfrac{2}{3}(1 - 2y)$ ⇒ 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.

Second equation : $\dfrac{2 + 5y}{3} = \dfrac{x}{7} - 2$

Step 1:

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

Let x = 0, then $\dfrac{2 + 5y}{3} = \dfrac{0}{7} - 2$ ⇒ y = -1.6

Let x = 7, then $\dfrac{2 + 5y}{3} = \dfrac{7}{7} - 2$ ⇒ y = -1

Let x = 14, then $\dfrac{2 + 5y}{3} = \dfrac{14}{7} - 2$ ⇒ y = -0.4

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.

On the same graph paper, draw the graph for each given equation.

Both the straight lines drawn meet at point P. As it is clear from the graph, co-ordinates of the common point are (7, -1).

Solution of the given equations is : x = 7 and y = -1.