Find the distance between the pair of parallel lines, y = x-11 & y = x-7

$$k:\ y = x-11\ \ \ \Leftrightarrow\ \ \ x-y-11=0\\ and\\ l:\ y = x-7\ \ \ \Leftrightarrow\ \ \ x-y-7=0\\\\the\ distance:\\\\ d(k; l)= \frac{|-11-(-7)|}{\sqrt{1^2+1^2} } =\frac{|-11+7|}{\sqrt{2} } =\frac{|-4|}{\sqrt{2} } =\frac{4\cdot \sqrt{2} }{\sqrt{2}\cdot \sqrt{2} } =\frac{4 \sqrt{2} }{2 } =2 \sqrt{2}$$

$$Given \ the \ equations \ of \ two \ non-vertical \ parallel \ lines:\\\\y = mx+b_1\\y = mx+b_2\\\\the \ distance \ between \ them \ can \ be \ expressed \ as : \\\\d= \frac{|b_{1}-b_{2}|}{ \sqrt{ m^2+1} }$$

$$y = x-11 \\ y = x-7 \\\\\\d= \frac{| -11- (-7)|}{ \sqrt{ 1^2+1} } =\frac{| -11+7|}{ \sqrt{ 1+1} } = \frac{|-4|}{ \sqrt{2} } = \frac{4}{ \sqrt{2} }\cdot \frac{\sqrt{2}}{\sqrt{2}}=\frac{4\sqrt{2}}{2}=2\sqrt{2}$$

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