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

x - 5y + 17 = 0 and 2x + y + 1 = 0

**Solution :**

x - 5y + 17 = 0 ----- (1)

2x + y + 1 = 0 ------(2)

(2) ⋅ 5 ==> 10x + 5y + 5 = 0 ----(3)

x - 5 y + 17 = 0

10 x + 5 y + 5 = 0

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

11 x + 22 = 0

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

11 x = - 22

x = -2

By substituting x = -2 in (1), we get

-2 - 5y + 17 = 0

15 - 5 y = 0

- 5y = - 15

-5y = - 15

y = 3

So the intersection point of the straight lines is (-2, 3).

**Example 2 :**

Find the intersection point of the straight lines

5x - 3y - 8 = 0 and 2x - 3y - 5 = 0

**Solution :**

5x - 3y - 8 = 0 ----- (1)

2x - 3y - 5 = 0 ------(2)

5 x - 3 y - 8 = 0

2 x - 3 y - 5 = 0

(-) (+) (+)

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

3 x - 3 = 0

x = 1

By applying x = 1 in the first equation

5(1) - 3y - 8 = 0

5 - 8 - 3y = 0

-3 - 3y = 0

-3y = 3

y = -1

So the intersection point of the straight lines is (1, -1).

**Example 3 :**

Find the intersection point of the straight lines

4x - 7y = 0 and 8x - y - 26 = 0

**Solution :**

4x - 7y = 0 ----- (1)

8x - y - 26 = 0 ------(2)

(2) ⋅ 7 => 56 x - 7 y - 182 = 0

4x - 7y + 0 = 0

56 x - 7y - 182 = 0

(-) (+) (+)

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

-52 x + 182 = 0

x = -182/(-52)

x = 7/2

By applying x = 7/2 in (1), we get

-4(7/2) - 7 y = 0

-14 - 7y = 0

-7y = 14

y = -2

So the point of intersection of the given lines is (7/2, -2).

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