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.

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> [!TLDR]
>
> Substitute $y=mx-3$ into $x-y=6$ to get $x(m-1)=-3$, i.e. $x=-\dfrac{3}{m-1}$. Since $m,x$ must both be integers, $m-1$ must be an integer divisor of $3$: $m-1\in\{1,-1,3,-3\}$.

Since the question asks to find **all** such lines, every integer divisor of the fixed number on the right must be checked — not just the ones that happen to look 'nice'.

**Step 1. Substitute the family into $x-y=6$ (i.e. $y=x-6$).**

$$mx-3=x-6$$
$$mx-x=-6+3$$
$$x(m-1)=-3$$
$$x=-\frac{3}{m-1}\qquad(m
e1)$$

**Step 2. Impose that both $m$ and $x$ are integers.**

$x$ is an integer exactly when $(m-1)$ divides $3$ exactly, i.e. $(m-1)$ is one of the four integer divisors of $3$: $\{1,-1,3,-3\}$. Each gives one value of $m$ (already an integer by construction) and one value of $x$:

| $m-1$ | $m$ | $x=-3/(m-1)$ | $y=mx-3$ | Line |
|---|---|---|---|---|
| $1$ | $2$ | $-3$ | $-9$ | $2x-y-3=0$ |
| $-1$ | $0$ | $3$ | $-3$ | $y+3=0$ |
| $3$ | $4$ | $-1$ | $-7$ | $4x-y-3=0$ |
| $-3$ | $-2$ | $1$ | $-5$ | $2x+y+3=0$ |

**Step 3. Verify each row on $x-y=6$.**

$(-3,-9)$: $-3-(-9)=6$ ✓  $(3,-3)$: $3-(-3)=6$ ✓  $(-1,-7)$: $-1-(-7)=6$ ✓  $(1,-5)$: $1-(-5)=6$ ✓ — all four intersection points genuinely lie on $x-y=6$, and each came from an integer $m$. …

Q.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.

Concept understanding — Family of Lines

$m-1$	$m$	$x=-3/(m-1)$	$y=mx-3$	Line
$1$	$2$	$-3$	$-9$	$2x-y-3=0$
$-1$	$0$	$3$	$-3$	$y+3=0$
$3$	$4$	$-1$	$-7$	$4x-y-3=0$
$-3$	$-2$	$1$	$-5$	$2x+y+3=0$

Q.Find all the equations of the straight lines in the family of the lines y=mx−3y = mx - 3y=mx−3, for which mmm and the xxx-coordinate of the point of intersection of the lines with x−y=6x - y = 6x−y=6 are integers.

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Q.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.

Unlock everything free for 14 days