EQUATION OF THE LINE PASSING THROUGH THE INTERSECTION OF THE LINES

Equation of the Line Passing Through the Intersection of the Lines :

Here we are going to see, how to find the equation of the line passing through the intersection of the lines

Equation of the Line Passing Through the Intersection of the Lines - Questions

Question 1 :

Find the equation of a straight line through the intersection of lines 7x + 3y = 10, 5x − 4y = 1 and parallel to the line 13x + 5y +12 = 0

Solution :

First, let us find the point of intersection of the given lines 7x + 3y = 10, 5x − 4y = 1.

7x + 3y = 10  ----(1)

5x − 4y = 1  ----(2)

4(1) + 3(2)

28x + 12y  =  40

15x - 12y  =  3

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

43x  =  43

x  =  1

By applying the value of x in (1), we get 

7(1) + 3y  =  10

7 + 3y  =  10

3y  =  10 - 7

3y  =  3

y  =  1

So, the point of intersection of the given lines is (1, 1)

The required line is passing through the point (1, 1) and parallel to the line 13x + 5y + 12  =  0

Slope of the required line  =  -13/5

(y - y1)  =  m(x - x1)

(y - 1)  =  (-13/5)(x - 1)

5(y - 1)  =  -13(x - 1)

5y - 5  =  -13x + 13

13x + 5y - 5 - 13  =  0

13x + 5y - 18  =  0

Question 2 :

Find the equation of a straight line through the intersection of lines 5x −6y = 2, 3x + 2y = 10 and perpendicular to the line 4x −7y +13 = 0

Solution :

5x −6y = 2 -----(1) 

3x + 2y = 10 -----(2) 

        (1)  ==>  5x −6y = 2 

     3(2) ==>   9x + 6y = 30

                ------------------

                    14x  =  32

x  =  32/14  =  16/7

By applying the value of x in (1), we get

5(16/7) - 6y  =  2

(80/7) - 6y  =  2

6y  =  (80/7) - 2

6y  =  (80 - 14)/7

y =  11/7

The point of intersection of the given lines is (16/7, 11/7)

The required line is perpendicular to the line 4x −7y +13 = 0

slope of the required line  =  -4/(-7)  =  4/7

(y - y1)  =  (-1/m)(x - x1)

(y - (11/7))  =  (-7/4)(x - (16/7))

(7y - 11)/7  =  (-7/4)(7x - 16)/7)

4(7y - 11)  =  -7(7x - 16)

28y - 44  =  -49x + 112

49x + 28y - 44 - 112  =  0

49x + 28y - 156  =  0


After having gone through the stuff given above, we hope that the students would have understood, "Equation of the Line Passing Through the Intersection of the Lines". 

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