# Prove that the lines d1, d2 are intercepting: (d1)14x-9y-24=0; (d2)7x-2y-17=0

neela | Student

(d1) 14x-9y-24=0.

(d2)  7x-2y-17=0.

We  write the equations  as below:

d1 : 14x - 9y = 24

d2 : 7x- 2y = 17.

Since  det |[(14  -9), (7 -2)]| = 14 *-2 - 7*(-9) = -28 +63 = 35 is not equal to zero, the  lines d1 and d2 must intersect at a point.

That proves that the lines are intercepting.

d1 - 2*d2  gives -9y-2(-2y) = 24- 2*17 = -10.

(-9 +4)y = -10.

-5y = -10.

y = -10/-5 = 2.

Put y = 2 in d2: 7x-2y =17 .

7x = 17+2y = 17 + 2*2 = 21.

7x = 21

x = 21/7 = 3.

The intercepting point has the coordinates: (3 , 2).

giorgiana1976 | Student

To prove that the lines d1 and d2 are intercepting we'll have to verify if the system formed from the equations of d1 and d2 has a solution.

We'll form the system:

14x-9y-24=0

14x - 9y = 24 (1)

7x-2y-17=0

7x - 2y = 17 (2)

We'll solve the system using elimination method. For this reason, we'll multiply (2) by -2 and we'll add the resulting equation to (1):

-14x + 4y = -34 (3)

(1) + (3): 14x - 9y - 14x + 4y = 24 - 34

We'll eliminate and combine like terms:

-5y = -10

We'll divide by -5:

y = 2

We'll substitute y in (1):

14x - 9y = 24

14x - 18 = 24

14x = 24 + 18

14x = 42

7x = 21

x = 3

The solution of the system represents the intercepting point of the lines. The intercepting point has the coordinates: (3,2).