Intersection of Two Straight Lines

Two lines will intersect at a point. The point will have a pair of values as (x1, y1). The straight lines are represented by equation with two or one variable in x and y.  If those equations are solved to get the value of x and y, will represent the point intersection of those two lines. To solve the set of lines to get the value of x and y , we can use either the method of elimination or substitution. Now let us discuss few problems on this topic intersection of two straight lines.

Example Problems on Intersection of Two Straight Lines

Ex 1: Find the point of intersection of the following two lines

2x + 3y = 10; 2x + y = 6.

Sol: Given: 2x + 3y = 10 --------------(1)

2x + y = 6  --------------(2)

The point intersection of above two lines can be obtained by solving them as follows:

(1) – (2) implies: 2x + 3y = 10

2x + y = 6

We get, 2y = 4

y = 2.

From (2), 2x + (2) = 6

2x = 6 – 2 = 4

2x = 4

x = 2.

Therefore, the point of intersection is (2, 2).

Ex 2: Find the point of intersection of the following two lines

x + 2y = 1; 5x + 4y = -7.

Sol: Given: x + 2y = 1 ------------------(1)

5x + 4y = -7 -----------------(2)

The point of intersection of above two lines can be obtained by solving the as follows:

(1) x 5 – (2) implies: 5x + 10y = 5

5x + 4y = -7

We get, 6y = 12

y = 2.

Therefore, from (1), x + 2(2) = 1

Implies, x = 1 – 4 = -3.

Therefore the point of intersection is (-3, 2).

Ex 3: Find the point of intersection of the following two lines

3x + y = 10; y = 7.

Sol: Given: 3x + y = 10 -------------(1)

y = 7 -----------(2)

Since, y = 7 is one of the line, the value of the y coordinate will be 7.

Therefore, from (1), we get , 3x + 7 = 10

3x = 10 – 7 = 3

x = 1.

Therefore, the point of intersection is ( 1, 7).

Practice Problems on Intersection of Two Straight Lines

1. Find the point of intersection of lines 3x + 2y = 7 and x + y = 3.

[ Answer: (1, 2)]

2. Find the point of intersection of lines 3x - 2y = -2 and x - 2y = 6.

[ Answer: (-4, -5)]

3. Find the point of intersection of lines 4x - 3y = -10 and 3x + y = -1.

[ Answer: (-1, 2)]