Verify if the lines are intercepting? x+y = 20 3x - 2y = 6

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neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

Posted on

The lines x+y = 20 and 3x-2y= 6 must inersect as their slopes are obviously looking different.

Slope of x+y = 20  or y = 20-x (which is in y = mx+c , m is slope ) is -1.

Slope of  of 3x-2y = 6 or -2y = -3x+6 ,  or y =(3/2)x- 6/2 , is 3/2.

Intersection of the lines are the solution of the equations:

x+y = 20..............(1)

3x-2y = 6..................(2)

Add 2*eq(1) and eq(2 ) . Then y gets eliminated:

2x+3x = 2*20+6 = 46

5x=46

x = 46/5= 9.2.

Eq(3)*3 - Eq(2) eliminates x:

3y- (-2y) = 3*20-6 = 56

5y = 56

y = 54/5 = 10.4

Therefore (x,y) = (9.2 , 10.4) are the coordinates of the point of intersection of the 2 lines.

william1941's profile pic

william1941 | College Teacher | (Level 3) Valedictorian

Posted on

It is quite simple to verify if two lines intersect. If the lines are not parallel, they intersect. For the lines given, we find the slope. This can be done by expressing them as y= mx+ b where m is the slope and b is the y- intercept. We only need to consider the slope.

  • x+y = 20

=> y = 20 - x

=> The slope of the line is -1.

  • 3x - 2y = 6

=> -2y = 6 - 3x

=> 2y = 3x - 6

=> y = (3/2)x -3

=> The slope of the line is 3/2

Therefore as the slopes of the lines are not the same the lines intersect.

Top Answer

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 formed form the equations of the lines.

x+y = 20 (1)

3x - 2y = 6  (2)

The solution of this system represents the coordinates of the intercepting point.

We'll solve the system using elimination method. For this reason, we'll multiply (1) by 2:

2x + 2y = 40 (3)

We'll add (3) to (2):

2x + 2y +  3x - 2y= 40 + 6

We'll eliminate like terms:

5x = 46

We'll divide by 5:

x = 46/5

We'll susbtitute x in (1) and we'll gte:

x+ y = 20

46/5 + y = 20

We'll subtract 46/5 both sides:

y = 2 - 46/5

y = (10-46)/5

y = -36/5

The lines are intercepting and the coordinates of the intercepting point are (46/5 , -36/5)

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