Verify if the lines x-3y-2=0 and 2x+y-5=0 are intersecting .

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The lines can be written in the slope intercept form y = mx + c

x- 3y - 2 = 0

=> 3y = x - 2

=> y = x/3 - 2/3

The slope is 1/3

2x + y - 5 = 0

=> y = -2x + 5

The slope is -2

As the slope of the two lines is not the same, they are not parallel, therefore they are intersecting as they don't have the same slope.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

To verify if the lines are intercepting, we'll have to solve the system fomed by the equations of the lines. If the system has solution, this solution represents the intercepting point of the lines.

We'll re-write the equation, moving the number to the right side:

x - 3y = 2 (1)

2x + y = 5 (2)

We'll multiply (2) by 3:

6x + 3y = 15 (3)

We'll add (1) and (3):

x - 3y + 6x + 3y = 2 + 15

We'll combine and eliminate like terms:

7x = 17

x = 17/7

We'll substitute x in (1):

17/7 - 3y = 2

We'll subtract 17/7 both sides:

-3y = 2 - 17/7

-3y = (14-17)/7

-3y = -3/7

We'll divide by -3:

y = 1/7

The lines are intercepting and the intercepting point is (17/7 ; 1/7).

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