Two equal sides of an isosceles triangle are given by 7x - y+ 3 = 0 and x + y- 3 = 0. If the slope of the third side is an integer , then m =

Slopes of the given lines are 7, - 1
Since these two lines are equal, the angles made by these lines with the third side in equal.

$\therefore \left|\frac{\mathrm{m}-7}{1+7\mathrm{m}}\right|=\left|\frac{\mathrm{m}+1}{1-\mathrm{m}}\right|\Rightarrow \frac{\mathrm{m}-7}{1+7\mathrm{m}}=\pm \frac{\mathrm{m}+1}{1-\mathrm{m}}$
$\Rightarrow$ (m-7)(1-m)=±(m+1)(1 +7m)
$\Rightarrow$ (-m² + 8m - 7)= ± (7m² + 8m+1)
$\Rightarrow$ 7m² +8m+1 = -m² + 8m - 7 or 7m² +8m+1  = m²-8m+7
$\Rightarrow$8m²=-8 or 6m² + 16m - 6 = 0 $\Rightarrow$m²=-1 or 3m²+8m-3=0
$\Rightarrow$ (m + 3)(3m- 1) = 0 $\Rightarrow$ m =- 3 or 1/3.