Video: Choosing the Correct Proportionality Statement Equivalent to a Given One

If 𝑦 ∝ (1/√𝑥) then which of the following is true [A] 𝑥 is inversely proportional to 𝑦² [B] 𝑥 is inversely proportional to 𝑦³ [C] 𝑥 is directly proportional to 𝑦² [D] 𝑥 is directly proportional to 𝑦 [E] 𝑥 is inversely proportional to 𝑦.

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Video Transcript

If 𝑦 is proportional to one divided by the square root of 𝑥, then which of the following is true? A) 𝑥 is inversely proportional to 𝑦 squared; B) 𝑥 is inversely proportional to 𝑦 cubed; C) 𝑥 is directly proportional to 𝑦 squared; D) 𝑥 is directly proportional to 𝑦; or E) 𝑥 is inversely proportional to 𝑦.

If we consider the question, 𝑦 is proportional to one divided by the square root of 𝑥 is the same as saying 𝑦 is inversely proportional to the square root of 𝑥. This will automatically rule out answer C and answer D, as they refer to direct proportion. If 𝑦 is inversely proportional to the square root of 𝑥, then 𝑥 must be inversely proportional when related to 𝑦.

We now need to alter the equation 𝑦 is proportional to one over square root of 𝑥 so that 𝑥 is the subject of the equation. Squaring both sides of the equation gives us 𝑦 squared is proportional to one divided by 𝑥. This is because the square root of 𝑥 multiplied by the square root of 𝑥 is 𝑥. Multiplying both sides of this equation by 𝑥 gives us 𝑥𝑦 squared is proportional to one. And finally, dividing both sides by 𝑦 squared gives us that 𝑥 is proportional to one divided by 𝑦 squared, or one over 𝑦 squared.

As 𝑥 is proportional to one over 𝑦 squared, we can say that 𝑥 is inversely proportional to 𝑦 squared. Answer A was true. If 𝑦 is proportional to one over the square root of 𝑥, then 𝑥 is inversely proportional to 𝑦 squared.

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