Verify if the line x=42-14y intersects the line y=2x-11.
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We have the line x = 42 - 14y. The other line is y = 2x - 11.
Use x = 42 - 14y from the equation of the first line and substitute in the second
y = 2x - 11
=> y = 2( 42 - 14y) - 11
=> y = 84 - 28y - 11
=> y = -28y + 73
=> 29y = 73
=> y = 73/29
x = 42 - 14 * ( 73/29)
=> x = 196/29
Therefore, the lines do intersect and the point of intersection is (196/29 , 73/29)
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To determine if the lines are intercepting each other, we'll have to solve the system formed form the equations of the lines and to see if it has a solution. The solution of the system represents the intercepting point of the lines.
We'll change the 1st equation in:
x+14y=42 (1)
We'll change the 2nd equation in:
2x-y=11 (2)
We'll solve the system using elimination method. For this reason, we'll multiply (2) by 14:
28x - 14y = 154 (3)
We'll add (3) to (1):
28x - 14y + x + 14y = 154 + 42
We'll eliminate like terms:
29x = 196
We'll divide by 29:
x = 196/29
We'll susbtitute x in (1) and we'll get:
196/29 + 14y=42
We'll subtract 196/29 both sides:
14y = 42 - 196/29
14y = (1218-196)/29
We'll divide by 14:
y = 1022/406
The lines are intercepting and the coordinates of the intercepting point are: (196/29, 1022/406).
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