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Given that π sub one of π₯ is equal to seven plus π over π₯ minus seven, π sub two of π₯ is equal to two over π₯ minus seven, and π sub one of π₯ is equal to π sub two of π₯, what is the value of π?
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Here, we have two rational functions.
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And weβre told that the two functions are equal to each other.
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So that means that for π sub one of π₯ to be called π sub two of π₯, the functions are equal for all values of π₯, not just for a certain value or values.
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So we can say that seven plus π over π₯ minus seven equals two over π₯ minus seven.
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Now, look really carefully.
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The denominator of each of our functions is equal.
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And so for the function π sub one of π₯ to be equal to the function π sub two of π₯, their numerator must also be equal.
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So seven plus π must be equal to two.
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Now, we have this; we can solve this equation for π fairly easily.
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Weβre going to subtract seven from both sides.
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And seven plus π minus seven is π.
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And two minus seven is negative five.
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And so we found the value of π; itβs negative five.
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Now, of course, we can check that what weβve done is correct by substituting π into our expression for the function π sub one of π₯.
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We get seven plus negative five over π₯ minus seven, which is two over π₯ minus seven as required.