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

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to convert the given form of equations of the lines into slope intercept form, such that:

`14x - 9y - 24 = 0 => -9y = -14x + 24 => y = 14/9 x - 8/3`

`7x - 2y - 17 = 0 => -2y = -7x + 17 => y = 7/2 x - 17/2`

You need to test if the given lines intersects each other, hence, you need to check if the same y coordinate holds for both equations, such that:

`14/9 x - 8/3 = 7/2 x - 17/2`

You need to bring the fractions ro a common denominator, such that:

`28x - 48 = 63x - 153 => 28 x - 63 x = 48 - 153`

`-35x = -105 => x = 3`

`y = 14/3 - 8/3 => y = 6/3 => y = 2`

Hence, testing if the lines intersects each other yields that they do at `(3,2).`

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

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).