In Exercises 43–46, let x represent one number and let y repre...

Question

In Exercises 43–46, let x represent one number and let y represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers. The difference between the squares of two numbers is 3. Twice the square of the first number increased by the square of the second number is 9. Find the numbers.

Accepted Answer

Welcome back. I am so glad you're here. We are given some information and we are to create a system of equations based on that information. Were to use X and Y for our variables. So we are told the squares of two numbers. So our two numbers are going to be X and Y squaring them means we're going to take x squared and y squared. Well, we're told that the difference of those two squares will take x squared minus y squared. The difference is equals five. Right? There's our first equation in our system of equations. Then we're told so we'll start our second equation that increasing three times the square of the first number, three times the square of the first number. So the first numbers X, the square of that number is x squared, three times the square of the first number is three X squared and then increasing it by the square of the other number. So we're adding, increasing is adding by the square of the other number. That's why squared again yields or equals 31. There's our system of equations and now we just use what we know from previous lessons on solving systems of equations and I'm going to use substitution here. I'm going to add y squared to both sides of this equation and then that new equation, one is going to read X squared equals five plus y squared. And so now anywhere that there is an X squared in the second equation, I can substitute in five plus Y squared. So I'll do that three times no longer X squared. But now five plus y squared plus y squared equals 31. So now we can do the work with this second equation and solve for why? Okay, so I'll distribute that 33 times five is 15, 3 times Y squared is plus three, Y squared plus Y squared equals 31 combining like terms of ry squares. And then I'll subtract 15 from both sides to get the constant terms on the right hand side. Three, y squared plus y squared is four. Y squared Equals 31 -15 is 16, Dividing both sides by four. And we get y squared equals 16 divided by four, is four. And now taking the square root of both sides And I get why equals plus or minus the square to four is 2. So there are my Y solutions. I could create a squiggly solutions set and I know to why solutions we've got why can be a negative two or why can be a positive two. And now I just have to figure out what my X values would be when y equals negative two and one Y equals positive two. And I'm going to use this first equation that I have over here where I got X squared by itself. And I'm going to use that equation To substitute -2 in for Y and to substitute positive to in for why? So bringing down that X squared equals five plus negative two squared will have X squared equals five plus. Well negative two squared is four, so five plus four. X squared equals five plus four. While x squared equals nine. And taking the square root of both sides of this equation, we get X equals the square of nine is plus or minus three. Well that's interesting. When Y equals negative two, we can have X equal a negative three, or X can also equal a positive three when y equals -2. So now we've got three possibilities. And now let's continue on plugging in just two for y. So that's X squared accelerated down here. X squared equals five plus two squared and x squared equals five plus two squared is four. So we have X squared equals nine. Again, taking the square root of both sides gives us X equals again plus or minus three. So when y equals two, X can equal a negative three or again why could equal or X could equal a positive three when Y equals positive two. So we've got four different possibilities here we look at our answer choices in this matches with answer choice D. Well done. We'll catch you on the next one

Key Concepts

Formulating Systems of Nonlinear Equations

Solving Systems of Nonlinear Equations

Properties of Squares and Differences of Squares

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