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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
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.
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.
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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