Video: AQA GCSE Mathematics Higher Tier Pack 4 β€’ Paper 3 β€’ Question 14

In the equation (𝑦π‘₯Β²/π‘˜) = 1, π‘˜ is a constant. Circle the correct statement. [A] 𝑦 is directly proportional to π‘₯Β² [B] 𝑦 is inversely proportional to π‘₯Β² [C] π‘₯ is directly proportional to 𝑦² [D] 𝑦 is inversely proportional to 1/π‘₯Β².

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Video Transcript

In the equation 𝑦π‘₯ squared over π‘˜ equals one, π‘˜ is a constant. Circle the correct statement. The options are 𝑦 is directly proportional to π‘₯ squared, 𝑦 is inversely proportional to π‘₯ squared, π‘₯ is directly proportional to 𝑦 squared, or 𝑦 is inversely proportional to one over π‘₯ squared.

Each of the four options describes a different type of proportional relationship between 𝑦 and π‘₯. So what we need to do in order to answer the question is determine how 𝑦 and π‘₯ are related to one another. In order to do this, we need to first rearrange the equation we’ve been given to give 𝑦 in terms of π‘₯.

The first step is to multiply both sides of the equation by π‘˜ to cancel out the π‘˜ in the denominator on the left-hand side. This gives 𝑦π‘₯ squared is equal to π‘˜. The next step is to divide both sides of the equation by π‘₯ squared, giving 𝑦 equals π‘˜ over π‘₯ squared. So we found 𝑦 in terms of π‘₯.

Now the fraction π‘˜ over π‘₯ squared can be written as π‘˜ multiplied by one over π‘₯ squared. So what we found is that 𝑦 is equal to a multiple of one over π‘₯ squared. That’s a multiple of the reciprocal of π‘₯ squared. We can express this using the proportionality symbol. 𝑦 is proportional to one over π‘₯ squared. Now as this is a reciprocal relationship, this means that as π‘₯ or π‘₯ squared increase, 𝑦 will decrease. So the two quantities 𝑦 and π‘₯ squared are inversely proportional to one another.

Be careful here because we’re given two possible answers involving inversely proportional relationships. The first is that 𝑦 is inversely proportional to π‘₯ squared, and the second is that 𝑦 is inversely proportional to one over π‘₯ squared. The correct answer is that 𝑦 is inversely proportional to π‘₯ squared because the inversely proportional part creates the reciprocal for us.

The statement 𝑦 is inversely proportional to one over π‘₯ squared would actually be written as 𝑦 is proportional to one over one over π‘₯ squared, which would actually mean that 𝑦 was proportional to π‘₯ squared because one over one over π‘₯ squared just simplifies to π‘₯ squared. So in this case, this would actually mean that 𝑦 was directly proportional to π‘₯ squared.

The correct answer found by rearranging the equation we were given to see that 𝑦 was equal to some multiple of one over π‘₯ squared is that 𝑦 is inversely proportional to π‘₯ squared.

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