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Question
Find all the equations of the straight lines in the family of the lines y = mx − 3, for which m and the x-coordinate of the point of intersection of the lines with x − y = 6 are integers
Solution
The equations of the given lines are
y = mx – 3 .......(1)
x – y = 6 .......(2)
Solving equations (1) and (2)
(2) ⇒ x – (mx – 3) = 6
x – mx + 3 = 6
x(1 – m) = 3
x = `3/(1 - "m")` .......(3)
From equation (3)
Let us find the values of x and m for which they are integers.
The only values of m for which, x is an integer are m = 0, 2, – 2
When m = 0
x = `3/(1 - 0)`
= 3
The corresponding equation is
y = 0 . x – 3
y + 3 = 0
When m = 2
x = `3/(1 - 2)`
= `3/(- 1)`
= – 3
The corresponding equation is y = – 2x + 3
2x + y – 3 = 0
When m = – 2
x = `3/(1 + 2)`
= `3/3`
= 1
The corresponding equation is
y = – 2 x + 3
2x + y – 3 = 0
∴ The required equations of the lines are
y + 3 = 0
2x – y – 3 = 0
and
2x + y – 3 = 0
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