# Intercepting lines. Prove that the lines are intercepting 2a-b +2=0  a  + b - 4 = 0.

sciencesolve | Certified Educator

The given lines intersects each other if the equations share a common solution, such that:

`{(2a - b + 2 = 0),(a + b - 4 = 0):}`

`{(2a - b = -2),(a + b = 4):}`

Performing the addition of equations yields:

`2a - b + a + b = -2 + 4 => 3a = 2 => a = 2/3`

`2/3 + b = 4 => b = 4 - 2/3 => b = (12 - 2)/3 => b = 10/3`

Hence, testing if the lines intersects each other yields that they do, at `a = 2/3, b = 10/3.`

giorgiana1976 | Student

We'll solve the system using substitution method. We'll change the 2nd equation into:

a+b = 4 => a = 4 - b (3)

We'll substitute (3) in (1):

2(4 - b) - b + 2 = 0

We'll remove the brackets:

8 - 2b - b = -2

We'll combine like terms and we'll subtract 8 both sides:

-3b = -2 - 8

-3b = -10

We'll divide by -3:

b = 10/3

We'll substitute b in (3):

a = 4 - 10/3

a = (12-10)/3

a = 2/3

The intercepting point of the lines is: (2/3 ; 10/3)