If 𝑥 squared plus nine 𝑦 squared equals 15 and 𝑥𝑦 equals two, what is the value of 𝑥 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 𝑥 squared plus nine 𝑦 squared equals 15 to see if it would give us something similar to 𝑥 minus three 𝑦 squared. However, 𝑥 squared plus nine 𝑦 squared equals 15 doesn’t fit any of our algebraic identities.
So what I’m gonna do is rewrite 𝑥 minus three 𝑦 squared as 𝑥 minus three 𝑦 times 𝑥 minus three 𝑦. We can expand and multiply here. 𝑥 times 𝑥 equals 𝑥 squared. 𝑥 times negative three 𝑦 equals negative three 𝑥𝑦. Negative three 𝑦 times 𝑥 equals negative three 𝑥𝑦. And negative three 𝑦 times negative three 𝑦 equals positive nine 𝑦 squared.
Negative three 𝑥𝑦 and negative three 𝑥𝑦 are like terms. We can combine them by adding their coefficients. Negative three plus negative three equals negative six. Bring down the 𝑥𝑦. And we found that 𝑥 minus three 𝑦 squared equals 𝑥 squared minus six 𝑥𝑦 plus nine 𝑦 squared. We know the value of 𝑥𝑦. And we know the value of 𝑥 squared plus nine 𝑦.
We can rearrange the equation to say 𝑥 squared plus nine 𝑦 squared minus six 𝑥𝑦 and then substitute what we know. We know that 𝑥 squared plus nine 𝑦 squared equals 15. We also know that 𝑥 times 𝑦 equals two. We now have 15 minus six times two. 15 minus 12 equals three.
𝑥 minus three 𝑦 squared is equal to three.