# HOW TO FIND POINT OF CONCURRENCY OF THREE LINES

(i)  Solve any two equations of the straight lines and obtain their point of intersection.

(ii)  Plug the coordinates of the point of intersection in the third equation.

(iii)  Check whether the third equation is satisfied

(iv)  If it is satisfied, the point lies on the third line and so the three straight lines are concurrent.

Example 1 :

Prove that the straight lines

2x+y  =  1, 2x+3y  =  3 and 3 x + 2 y = 2

are concurrent. Find the point of concurrency.

Solution :

The given equations are

2x + y  =  1     -----(1)

2x + 3y  =  3   -----(2)

3x + 2y  =  2   -----(3)

Multiply the 1st equation by 3 and subtract the 2nd equation from 1st equation.

3(2x+y) - (2x+3y)  =  3(1) - 3

6x+3y-2x-3y  =  3 - 3

4x  =  0

Divide 4 on both sides

4x/4  =  0/4

x  =  0

Apply x = 0 in the 1st equation, we get

2(0) + y = 1

y = 1

Now let us apply the point (0, 1) in the third equation

(3)===>   3 x + 2 y = 2

Here x = 0 and y = 1

3 (0) + 2 (1)  =  2

0 + 2  =  2

2  =  2

Since the point (0, 1) satisfies the 3rd equation, we may decide that the point(0, 1) lies on the 3rd line.

Hence the given lines are concurrent and the point of concurrency is (0, 1).

Example 2 :

Prove that the straight lines

3 x + y = -2, 2 x - y = -3, x + 4 y = 3

are concurrent. Find the point of concurrency.

Solution :

The given equations are

3x+y = -2     -----(1)

2x - y = -3  -----(2)

x + 4y = 3   -----(3)

Adding 1st and 2nd equation we get,

3x + y  + 2x - y  =  -3 + (-2)

5x  =  -5

Divide by 5 on both sides

5x/5  =  -5/5

x  =  -1

Apply x = -1 in the 1st equation, we get

3(-1) + y = -2

-3 + y = -2

Add 3 on both sides we get,

-3 + 2 + y  =  -2 + 3

y  =  1

Now let us apply the point (-1, 1) in the third equation

(3)===>  x + 4 y = 3

Here x = -1 and y = 1

-1 + 4 (1)  =  3

-1 + 4  =  3

3  =  3

Since the point (-1, 1) satisfies the 3rd equation, we may decide that the point(-1, 1) lies on the 3rd line.

Hence the given lines are concurrent and the point of concurrency is (0, 1). If you need any other stuff in math, please use our google custom search here.

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