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Question
If a : b : : c : d, then prove that
(ax+ by): (cx + dy)=(ax - by) : (cx - dy)
Solution
(ax+ by): (cx + dy)=(ax - by) : (cx - dy)
`"a"/"b" = "c"/"d"`
Multiplying both sides by `"x"/"y"`
`=> "a"/"b" xx "x"/"y" = "c"/"d" xx "x"/"y"`
`=> ("ax")/("by") = ("cx")/("dy")`
Applying componendo and dividendo,
`("ax+ by")/("ax - by") = ("cx + dy")/("cx - dy")`
` => ("ax+ by")/("cx + dy") = ("ax - by")/("cx - dy")`
Hence, ax + by : cx + dy = ax - by : cx - dy
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