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

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neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

Posted on

(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's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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

We'll add 24 both sides:

14x - 9y = 24 (1)

7x-2y-17=0

We'll add 17 both sides:

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).

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