Video: Evaluating Algebraic Expressions Using Algebraic Identities

If π‘₯Β² + 9𝑦² = 15 and π‘₯𝑦 = 2, what is the value of (π‘₯ βˆ’ 3y)Β²?

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Video Transcript

If π‘₯ squared plus nine 𝑦 squared equals 15 and π‘₯𝑦 equals two, what is the value of π‘₯ minus three 𝑦 squared?

At first, it might not be really clear what the best path for solving this is. You might want to factor π‘₯ squared plus nine 𝑦 squared equals 15 to see if it would give us something similar to π‘₯ minus three 𝑦 squared. However, π‘₯ squared plus nine 𝑦 squared equals 15 doesn’t fit any of our algebraic identities.

So what I’m gonna do is rewrite π‘₯ minus three 𝑦 squared as π‘₯ minus three 𝑦 times π‘₯ minus three 𝑦. We can expand and multiply here. π‘₯ times π‘₯ equals π‘₯ squared. π‘₯ times negative three 𝑦 equals negative three π‘₯𝑦. Negative three 𝑦 times π‘₯ equals negative three π‘₯𝑦. And negative three 𝑦 times negative three 𝑦 equals positive nine 𝑦 squared.

Negative three π‘₯𝑦 and negative three π‘₯𝑦 are like terms. We can combine them by adding their coefficients. Negative three plus negative three equals negative six. Bring down the π‘₯𝑦. And we found that π‘₯ minus three 𝑦 squared equals π‘₯ squared minus six π‘₯𝑦 plus nine 𝑦 squared. We know the value of π‘₯𝑦. And we know the value of π‘₯ squared plus nine 𝑦.

We can rearrange the equation to say π‘₯ squared plus nine 𝑦 squared minus six π‘₯𝑦 and then substitute what we know. We know that π‘₯ squared plus nine 𝑦 squared equals 15. We also know that π‘₯ times 𝑦 equals two. We now have 15 minus six times two. 15 minus 12 equals three.

π‘₯ minus three 𝑦 squared is equal to three.

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