EXAMPLES OF FINDING THE POINT OF INTERSECTION OF TWO LINES

If two straight lines are not parallel then they will meet at a point.This common point for both straight lines is called the point of intersection.

If the equations of two intersecting straight lines are given,then their intersecting point is obtained by solving equations simultaneously.

Example 1 :

Find the intersection point of the straight lines

4x - 3y  =  3 and 3x + 2y  =  15

Solution :

4x - 3y  =  3   ----- (1)

3x + 2y  =  15 ------(2) 

(1) ⋅ 2 =>  8x - 6y  =  6

(2) ⋅ 3 => 9 x + 6 y  =  45

                          8 x - 6 y  =  6

                          9 x + 6 y  =  45

                       --------------------

                          17x  =  51 

                          x  =  51/17

                          x  =  3

By applying x = 3 in (1), we get

8(3) - 6y  =  6

24 - 6y  =  6

-6y  =  6 - 24

-6 y  =  -18

y  =  3

So the point of intersection of the straight lines is (3, 3).

Example 2 :

Find the intersection point of the straight lines

3x + 2y  =  11 and 7x - 3y  =  41

Solution :

3x + 2y  =  11 ----- (1)

7x - 3y  =  41  ------(2) 

(1) ⋅ 3 =>  9x + 6y  =  33

(2) ⋅ 2 =>  14 x - 6 y  =  82

9x + 6y  =  33

14x - 6y  =  82

--------------------

23x  =  115 

x  =  115/23

x  =  5

By applying x = 5 in (1), we get

3(5) + 2y  =  11

15 + 2y  =  11

y  =  11 - 15

2y  =  -4

y  =  -2

So the point of intersection of the given straight lines is (5, -2).

Example 3 :

Find the intersection point of the straight lines

5x + 3y  =  11 and 3x + 5y  =  -3

Solution :

5x + 3y  =  11  ----- (1)

3x + 5y  =  -3 ------(2) 

(1) ⋅ 5 => 25 x + 15 y = 55

(2) ⋅ 3 => 9 x + 15 y = -9

25 x + 15 y = 55

9 x + 15 y = -9

(-)     (-)    (+)

--------------------

16x  =  64 

x  =  4

By applying x = 4 in (1), we get

5(4) + 3y  =  11

20 + 3y  =  11

3y  =  11 - 20

3y  =  -9

y = -3

So the intersection point of the straight lines is (4,-3).

Example 4 :

Find the intersection point of the straight lines

2x - y  =  15 and 5x + 3y  =  21

Solution :

2x - y  =  15   ----- (1)

5x + 3y  =  21 ------(2) 

(1) ⋅ 3 => 6x - 3y  =  45

6x - 3y  =  45

5x + 3y  =  21

--------------------

11x  =  66 

x  =  6

By applying x = 6 in (1), we get

2(6) - y  =  15

12 -  y  =  15

y  =  -3

So the point of intersection of the given straight lines is (6, -3).

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