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

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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).` **

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).