Question

Draw the graph of the equations x + y = 3, 2x – y = 3 and x + 2y = 4. Show that these three lines pass through the same point. Find the co-ordinates of this common point.

Answer 1

To determine if the given three lines pass through the same point, we need to find the intersection point of any two lines and then verify if it lies on the third line.

1. Solving the system of equations

We have the following linear equations:

1. x + y = 3
2. 2x – y = 3
3. x + 2y = 4

Step 1: Find the intersection of equations (1) and (2)

Add equations (1) and (2):

(x + y) + (2x – y) = 3 + 3

3x = 6

⇒ x = 2

Substitute x = 2 back into equation (1):

2 + y = 3

⇒ y = 1

The point of intersection for the first two lines is (2, 1).

Step 2: Verify the point on equation (3)

Substitute x = 2 and y = 1 into equation (3):

x + 2y = 2 + 2(1)

= 2 + 2

= 4

Since the left-hand side equals the right-hand side (4 = 4), the point (2, 1) lies on the third line as well.

2. Graphical Representation

The graph below illustrates the three lines and their common intersection point.

3. The lines x + y = 3, 2x – y = 3 and x + 2y = 4 are concurrent, meaning they all pass through a single common point.

The co-ordinates of the common point are (2, 1).