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Question
Find atleast two equations of the straight lines in the family of the lines y = 5x + b, for which b and the x-coordinate of the point of intersection of the lines with 3x − 4y = 6 are integers
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Solution
The equations of the given straight lines are
y = 5x + b .......(1)
3x – 4y = 6 .......(2)
To find atleast two equations from the family y = 5x + b for which b is an integer and x-coordinate of the point of intersection of (1) and (2) is an integer.
Solving (1) and (2) using equation (1) inequation (2) (2)
⇒ 3x – 4(5x + b) = 6
3x – 20x – 4b = 6
– 17x = 6 + 4b
x = `(6 + 4"b")/(- 17)`
When b = 7
We have x = `(6 + 28)/(- 17)`
= `34/(- 17)`
= – 2
The corresponding equation of the line is = 5x + 7
When b = – 10
We have x = `(6 - 40)/(- 17)`
= `(- 34)/(- 17)`
= 2
The corresponding equation of the line is y = 5x – 10
Thus y = 5x + 7 and y = 5x – 10 are the two straight lines belonging to the family such that b is an integer and the x-coordinate of the point of intersection with the line (2) is an integer.
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