What is the Point of Intersection of Two Lines Formula?

Point of Intersection of Two Lines Formula: The point of intersection is the point where two lines or two curves meet each other. The point of intersection of two lines or two curves is a point. If two planes meet each other then the point of intersection is a line.

It is defined as the common point of both the lines or curves that satisfy both the curves which can be derived by solving the equation of the curves.

The point of intersection formula is used to determine the meeting point of two lines. These lines can be represented by the equations a₁x + b₁y +c₁ = 0 and a₂x + b₂y + c₂ = 0. Additionally, it is possible to find the intersection point for three or more lines.

Point of Intersection of Two Lines Formula

If we consider two lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 the point of intersection of these two lines is given by:

Point of Intersection (x, y) = ((b₁×c₂ − b₂×c₁)/(a₁×b₂ − a₂×b₁), (c₁×a₂ − c₂×a₁)/(a₁×b₂ − a₂×b₁))

Point of intersection

Derivation of the point of intersection of two lines: 

Given equations:

→ a1x + b1y + c1 = 0 -> eq-1

→ a2x + b2y + c2 = 0 -> eq-2

Solving the equations using cross multiplication method:

       x     y     1

    b1    c1    a1    b1

    b2    c2    a2    b2

On cross-multiplying the constants we obtain:

→ x/(b1*c2 – b2* c1) = y/(c1*a2-c2*a1) = 1/(a1*b2-a2*b1)

Solving for x:

→ x/(b1*c2 – b2* c1) = 1/(a1*b2-a2*b1) 

→ x = (b1*c2 – b2* c1)/(a1*b2-a2*b1)

Solving for y:

→ y/(c1*a2-c2*a1) = 1/(a1*b2-a2*b1)

→ y=(c1*a2−c2*a1)/(a1*b2−a2*b1)

Hence point of intersection:

(x,y) = ((b₁×c₂ − b₂×c₁)/(a₁×b₂ − a₂×b₁), (c₁×a₂ − c₂×a₁)/(a₁×b₂ − a₂×b₁))

If two lines are parallel they never intersect each other:

Condition for two lines a1x + b1y + c1 = 0, a2x + b2y + c2 = 0 to be parallel 

 a1/b1 = a2/b2. 

Sample Problems on Point of Intersection of Two Lines Formula

Question 1: Find the point of intersection of line 3x + 4y + 5 = 0, 2x + 5y +7 = 0.

Solution:

The point of intersection of two lines is given by :

 (x, y) = ((b1*c2−b2*c1)/(a1*b2−a2*b1), (c1*a2−c2*a1)/(a1*b2−a2*b1))

 a1 = 3, b1 = 4, c1 = 5

 a2 = 2, b2 = 5, c2 = 7

 (x,y) = ((28-25)/(15-8), (10-21)/(15-8))

 (x,y) = (3/7,-11/7)

Question 2: Find the point of intersection of line 9x + 3y + 3 = 0, 4x + 5y + 6 = 0.

Solution:

The point of intersection of two lines is given by :

 (x,y) = ((b1*c2−b2*c1)/(a1*b2−a2*b1), (c1*a2−c2*a1)/(a1*b2−a2*b1))

 a1 = 9, b1 = 3, c1 = 3

 a2 = 4, b2 = 5, c2 = 6

 (x, y) = ((18-15)/(45-15), (54-12)/(45-15))

 (x, y) = (1/10, 7/5)

Question 3: Check if the two lines are parallel or not  2x + 4y + 6 = 0, 4x + 8y + 6 = 0

Solution:

To check whether the lines are parallel or not we need to check a1/b1 = a2/b2

a1 = 2, b1 = 4

a2 = 4, b2 = 8

2/4 = 4/8

1/2 = 1/2

Since the condition is satisfied the lines are parallel and can’t intersect each other.

Question 4: Check if the two lines are parallel or not  3x + 4y + 8 = 0, 4x + 8y + 6 = 0

Solution:

To check whether the lines are parallel or not we need to check a1/b1 = a2/b2

a1 = 3, b1 = 4

a2 = 4, b2 = 8

3/4 is not equal to 4/8

Since the condition is not satisfied the lines are not parallel.

Question 5: Check whether the point (3, 5) is point of intersection of lines 2x + 3y – 21 = 0, x + 2y – 13 = 0.

Solution:

A point to be a point of intersection it should satisfy both the lines.

Substituting (x,y) = (3,5) in both the lines

Check for equation 1: 2*3 + 3*5 – 21 =0 —-> satisfied

Check for equation 2: 3 + 2* 5 -13 =0 —-> satisfied

Since both the equations are satisfied it is a point of intersection of both the lines.

Question 6: Check whether the point (2, 5) is point of intersection of lines x + 3y – 17 = 0, x + y – 13 = 0

Solution:

A point to be a point of intersection it should satisfy both the lines.

Substituting (x,y) = (2,5) in both the lines

Check for equation 1: 2+ 3*5 – 17 =0 —-> satisfied

Check for equation 2: 7 -13 = -6  —>not satisfied

Since both the equations are not satisfied it is not a point of intersection of both the lines.

Question 7: Find the point of intersection of lines x = -2 and 3x + y + 4 = 0

Solution:

On substituting x = -2 in 3x + y + 4 = 0

-6 + y + 4 = 0;

y = 2;

So the point of intersection is (x,y) = (-2,2)                   

Practice Problems on Point of Intersection of Two Lines Formula

1. Find the point of intersection of the lines represented by the equations: 2x + 3y – 6 = 0 and 4x − y + 8 = 0.

2. Determine the point of intersection for the following pair of lines: 5x − y − 4 = 0 and 3x + 2y − 7 = 0.

3. Calculate the intersection point of these lines: x − 2y + 1 = 0 and 2x + y − 5 = 0.

4. Find the point of intersection of the given lines: 3x + 4y − 12 = 0 and 6x − y + 2 = 0.