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Slope of a Line Connecting Two Points

Show that the...

Question

Show that the line given by $x (a + 2 b) + y (a + 3 b) = a + b$ for different values of $a$ and $b$ pass through a fixed point.

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Solution

$\begin{matrix} x (a + z b) + y (a + z b) = a + b a (x + y) + b (2 x + 3 y) = a + b x + y = 1 \to (i) 2 x + 3 y = 1 \to (i i) M u l t i p l y 2 i n e q . (i) 2 x + 2 y = 2 2 x + 3 y = 1 - - - ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ - y = 1 y = - 1 p u t y i n e q (i) x - 1 = 1 x = 2 f i x e d p o i n t (2, - 1) \end{matrix}$

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